A Treatise on Fluxions ...

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1

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TREATISE

OK

F LU X I O JV S.

IN TWO VOLUMES.

• - •

BT

COLIN MACLAVRm, A M

I-t. P,»fa« rf M.a«™ti„ ii the U„lr«ri.y of Eainb«,k. «ij Fdlow of tie

Royal Society.

SECOND EDITION.

TO WBICH It njtFIXIO

AN JCCOUNT OF HIS LIFE.

TM WHOLE CAMroiAT eOMMMCm JM »tTlllD BY

Jn Eminent Mathematician.

ILLUSTRATED WITH FORTY^NE COPPERPLATES.

VOL. I.

LONDON:

MINTED FOE WILLIAM BAYNES, 54, FATERKOSTER ROW

AND WILLIAM DAVIS. '

■dhor of the Otatlanaa*! Majhoaatieal CqnipaaJoii, Author rf a. n t »^ ^ *

tOLD ALSO BY

1801.

Xnlght & Comptob, PrintMi, Middk SttMt,
Cloth Fjdr.

K

\ »

Wl

Mathi-Econ.
Ubnuy

ADVERTISEMENT.

^^HEN it is considered that the present Subject
is of a Nature the most exalted that the human
Mind ever conceived — that it is the most noble
invention, antient or modem, that ever was ob«
tained by the researches of Man — that its Inventor,
Sir Isaac Newton, by it has immortalized our
Country, and, at the same time, has given a
Clew to obtain Truths of- so sublime a nature as
no other method of reasoning could lead to;— «.
that this Work is the most elementary and
compleat of any ever written — that it has been
long out of print, and almost unattainable from
its Scarcity and price — the Publishers conceive,
that, for what they have here done, an Apology is
unnecessary.

The Editor.

Apnl 14, 1801.

•r\

X

0'

333991

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r

^ P'' ' J r

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k

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to
AIS GRACE

T •

<

Till

DUKE OP ARGYUE

▲RD

GREENWICH*

Xt is uot the shining figtO^ yaur GftACE has
made in the highest 'Employmttits^ neither is it
&e Lustre which your quitting them has added
to your Character ; it is not the Favour of
Princes, which you have oft6n enjoyed, nor
the loud and universal Applauses of the People,
which you at present possess ; but it is the
steady Virtue which has conducted you through-
out, that determines me to offer this Work
to your Gbac£^. And, having been honoured
with the Countenance and Favour of so great

b a Man,

\

VI

D£BICATION.

a Man, I embrace the Opportunity of express*
ing my Gratitude at a Time when I can have
no other Motive than that I am, with Truth,
and th^ utmost Respect^,

r Mt LoaD,

Your Grace's much obliged.
Most faithful,
And most humble Servaot^

Colin Macjlaurxn.

TVLZTACt.

PREFACE.

aiaatHMM

■fe

A LETTER published k the year 1784, uniler the
title of 7%e Afialystj first gave occasion to the
ensuihg Treatise ; and several reasons concurredl to
induce me to write on this Subject at. so great a
length. The Author of that Pieee had represented
the Method of Fluxions as founded on ialse Reason*
ing, and full of Mysteries. His Olbgections seemed
to have been occasioned^ in a giMt measure, by the
cdncise manner in which the Elements of this
.Method have been usually described ; and their
having been so much misunderstood by a person of
his abilities, appeared to me a sufficient proof that
a &lle(r Account of the Grounds of them was
requisite*

Though there can be no fcoMparison made l)etwixt
the extent or usefulness of the antient and modem
Disco verits in Geometry, yet it seems to be ge*
nerally allowed that the! Antients took greater care,
and were more succesafid in preserving the Cha*
racter of its Evidence enjtire. This determined me^
immediately after that Piece came to my hands,
and before I knew any .thing of what was intended

bS by

viii 7iitYAe)B.

by others in answer to it, to attempt to deduce
those Elements after the manner of the Antients,
from a few unexceptionable principles, by Demon-
strations of the strictest form. In my first Essay of
this kind, I contented lriy«lf ^th demonstrating
the principal Cases of the Propositions of the four
first Chapters of the first Book, and of the first
Chapter of the necmti Bmk ^ «iie folk^^ng Trea-
tise, nearly in the same form in which they now
appear. But when it was communicated to soipe
gentteaien, Ihey exprei^seil ot qfeMte that ttus kaMte
Method 'Vff Dcmomtmuitm shtnsli ke n^&rided .M
othir iMiascbes of Dhi^ Tteir)ty, axnitknt l^wba^id
cniavge die Plan. While I prooMtrdili tin Wiobki
t percdved that MtHe S&alH ^mie fdeftttive 1* iti^
aecuY^; H^at tlie Aesoiatioa | 'of Mvkral IMhilemt
which teoi been ddkteed in ^t nkjrs'terlMS liinaiixer,
by seciotfd AttiUfaM FhiKioiM» 'oesid be cbnpltted
with ^eftl»r 'eviAen)d9, and less 'danger of emit, by
first FtuKions ohly ; and tint oAm ProbldKit iUhfl
Wen resoltied by i^proatimfltioliB^ whRi lain 4ecunGtt
Solution could be oblHiiMd^itiiitbe<pame<*rygteaMr
Ibcility. l^se, ^(niAi €ftlMi ^obunrfltions ^onbelving
this method, and its application, led me on gmdwaUy
to compose a Treatise of a much greater extent than
I intended, dr wmild iiwft ttgtged da, if I had
been awiu« ijfit whm 1 begsn fhift Wo4c, teca»«
my aittendsDce in Nihe IJiurcnity «duld t^km me to
bestow but ti maXi fwnt 4>f .my time 'in cfeurying it
cm. And M lids hte been <Sie occnsiMi of my ddoy
in publMiing it, go I h«pe it waH berve for «n
spology^ ^f ^^^ tnlAafces bave >e«c^Bped me an
treatmg of sUeh ft VMtekyofvubjdOto, in a naifnisr

difierent

/

diiereiil; iMm tbajl ia lAkk tfaoy h«ve iMO HiiMUy
explained

In tfic iMaa tim^ liha Dofraoe of tbe Medwd of
]%ixioiis> and of tibe gcwt l&yenlor, ym x«ot meg-
iected. Besides an answer to The Jmaly^t that
appaared very early uiidtr dia nama oSJMtHahthat
dmUabrigivm^ (^ tl^ Au^khor kad ooncttled his
real naose aa the AmUyst vhem he oppoaed had done),
aaeeond, bjr^esaaubaad, in Itefenca of the tot, a
Diaooufse by Mm. 2^hi»i, a Traatiso of Sir Isaac Newr
*am% irith a Commentaiy by Mr. Cidmn^ and save'
r»l ofthev Pieces, waro pubHshad on this Subject
After I saw tiiat 9a asuch had been written upon it
ta so good a purpoaa^ I waa the la^r induced to
delay the piibBoation of this Treatise, till I could
finiah my desig». I accommodated my Definition
of the Variation of Curvature in Chap. xL to Sir
JTm^c Nnftm\ to prevent mistakes, as I have db^
served in Article 38^, but made no material aher<*
9tion in any diing eke. The greatest part of the
fiiat Book was printed in 17S7: but it could not
liave been ae useful to the Reader without the
second ; and I would recommend to him (if he ig
not already acquainted wi& this method), to
peruse ti|6 two first Chapters of the second Book,
before the five last of the first ; there being a few
passages in these which I couM not well avoid, that
will be better understood by one who has some
knowledge of the principal Rules of the Method of
Computation d^Uver^ in the second Book^

b3 In

X^ 9EEVACE.

In explaming the Nodon of & Flttxtott, I hava
followed Sir Udat Newton in the first Book,^
imagining that there can be no difficulty in coil*
cjciving Velocity wherever there it Motion ; nor do
I think that I have departed from his Sense in the
second Book ; and in both I have endeavoured to
avoid several expressions, which, though convenient,
might be liable to exceptions, and, perhaps, occasion
disputes. I have always represented Fluxions of all
all Ordeits by finite Quantities, the Supposition of an
infinitely little Magnitude being too bold a Postu^
latum for such, a Science as Geometry. But, bcr
cause the Method of Infiniteftimals. is much in use,
and is valued for its conciseness, I thought it was
requisite to acicoimt explicitly for the truth, and
perfect accuracy of the conclusions that are derived
from it ; the rather, that it does not seem to be a
very proper reason tliat is assigned by Authors, when
they determine what is called the Diference (but
more accurately the Fluxion) of a Quantity, and
tell us, That they reject certain Parts of the Element,
because they become infinitely less than the other
parts ; not only because a proof of this nature teay
leave some doubt as to the accuracy of the concluv
sion, but because it may be demonstrated that those
parts ought to be. neglected by them at any ratc^ or
that it would be an enor to retain them. If an Ac?
countant, that pretends to a scrupulous exactness,
should tell us that he had neglected certain Articles,
because he found them to be of small importance,
and it should appear that they ought not to have
been taken into consideration by him on that occa-
sion, but belong to a difierent account, we

shoul4

PkEFACE. Xi

tshoold approve lik conclusions as accurate, but

not his reason. Tliis method, however, OHty be con-
sidered as to easy and ready vay^ of distinguishing

what Parts of an Element are to be rejected, and

which are to be retained, in determining the precise

Fluxion of a Quantity, or the rate according to

which it increases or decreases.

^ After I found that this Treatise could not be conr
veniently contained in one volume, I was obliged
to reprint two leaves (pages 41 1, &c.), that it might
be divided into two. I have rq)rinted likewise the
first sheet, chiefly on account of several mors of
the press that had got into it, and one other leaf
(page S44), for the sake of a Pasnge, the omission
of which possibly would have been misinterpreted.
There are some Demonstrations in the first Chapter
of the first Book that might have been abridgedt
and some, perhaps, will appear unnecessary. I have
mentioned the reasons that induced me to insist so
fully on those Elementary Parts, in Articles 69, 104,
494, and 697.

t

Several Treatises have appeared while this was in
the press, wherein some of the same Problems have
been considered, though generally in a difierent
manner. I have had occasion to mention most
of them in the last Chi^ter of the second Book ;
but had not tha^e an opportunity to take notice^
that the Problem in 480 has been considered by

^ ADudnftotlie fot Editioii by tbe Author*

b4 Mr.

Mf* Jfiutfr ia his Mecbanic& I?l most of the iA*
staac4(8 wbar^ f&y conclusions did not* agree with
tiioms given by other Authors, I have pot mentioned
tfasir nynes*

(/.Ifi upon the Tiiiolc, the Evidence of this method
be represented to the satisfaction of th^ Reader^
some of the abstruse parts illustrated, or any imr
proyenlents of this useful Art be proposed, I shall
be under no great concern, though exceptions may
be made to some modes of Expression, or to sud)
Passages of this Treatise as are not ^sentia) to. tb#
|)rincipal design^

UFX

LIFE OF THE AUTHOR.

JSrLlSL. MACLAITRIN^ a most emitient mathemap
tician and philotapber, vas the son of a clergyman,
and bom at Kilmoddan, in Scotland, in the year
1698* He was sent to the University of Glasgow
in 17Q9;^here he continued five years, and applied
to Ins studio in a very intense manner, and particu^*
lariy to the inathematics. His great genius for
inathematical learning discovered itself so early as
at twelve years of age ; when, having accidentally
met with a copy of Euclid's Elements in a friend's
chamber, he became in a few days itHster of the first
6 books without any assistance : and it is certain,
that in his 1 6th year he had invented many of the
propositions which were afterwards published as part
^f his work entitled Geometria Organica. In his
15th year he took the degree of Master of Arts ; on
which occasion he composed and publicly defended
a Thesis on the Power of Gravity, with great ap-
plause. After this he quitted the University, and
retired to a country seat of his uncle, who had the
care of his education ; his parcruts being dead some
time. Here he spent two or three years in pursuing
his favourite studies; but, in 1717, at 19 years of
■ge only, he offered himself a candidate for the Pro-
fessorship of Mathematics in tlie IVIarischal College
of Aberdeen, and obtained it after a ten days' trial
against a very able competitor.

In 1719r Mr. Maclaurin visited London, where
he left his Geometria Organica to print, and where

he.

Xiv LIFE OF'THE AUTHOfi.

he became acquainted with Dr. Hoadley, then Bishop
of Bangor, Dr. Clarke, Sir Isaac Newton, and other
eminent men ; at which time also he was admitted a
member of the Royal Society: and in another
journey, in 1721, he contracted an intimacy with
Martin Folkes, Esq, the President of it> which con-
tinued during his whole life.

In 172S, Lord Polwarth, Plenipotentiary of the
King- of Great Britain at the Congress of Cambray,
engaged Maclaurin to go as a tutor and companion
to his eldest son, who was then to set out on his
travels. After a short stay at Paris, and visiting
other towns in France, they fixed in Lorrain; where
he wrote his piece. On the Percussion of Bodies^
which gained him the prize of the Royal Academy
of Sciences for the year 1724. But his pupil dying'
' soon after at Montpelier, he returned immediately
to his profession at Aberdeen. He was hardly
settled here, when he received an invitation to
Edinburgh ; the curators of that University being
desirous that he should supply the place of Mr.
Jkmes Gregory, whose great age an^ infirmities had
rendered him incapable of teaching. He had here
some difficulties to encounter, arising from competi*>
tors,^ who had good interest with the patrons of the
University, and also from the want of an additional
fund for the new professor ; which, however, at length
were all surmmounted, principally by the means
of Sir Isaac Newton. Accordingly, in Nov. 1725,
he was introduced into the University ; as was at the
same time his learned colleague and intimate friend
Dr. Alexander Monro, Professor of Anatomy. After
this, the mathematical classes soon became very

numerousi

LITE OP THE AXTTHOB. XT

BtniierouS)' there being generally upwards of 100
ftudents attending his Lectures every year; who
beiBg of difierent standings and proficiency, he was
obliged to divide them into four or five classes, ia
€ach of which he employed a full hour every- day
from the first of November to the first of June. In
the first class he taught the first 6 books of Euclid's
JELlements, Plane Trigonometry, Practical Geometry,
the Elements of Fortification, and an Introduction
to Algebra. The second class studied Algebra,
with the 11th and ISth books of Euclid, Spherical
Trigonometry, Conic Sections, and the General
Principles of Astronomy, The third went on in
Astronomy and Perspective, read a part of Newton's
Principia, and had performed a course of experi*
ments for illustrating them : he afterwards read and
demonstrated the Elements of Fluxions. Those in
the fourth class read a System of Fluxions, the
JDoctrine of Chances, and the remainder of Newton's
Principia.

In 1734, Dr. Berkley, Bishop of Cloyne, pub-
lished a piece called The Analyst ; in which he took
occasion, from some disputes that had arisen con*
ceming the grounds of the fluxionary method, to
explode the method itself; and also to charge
pwthematicians in general with infidelity in religion.
Maclaurin thought himself included in this charge,
and began an answer to Berkley's book : but other
answers coming out, and, as he proceeded, so many
discoveries, so many new theories and problems oc-
curred to him, that, instead of a vindicatory pamph^i
he produced a Complete System of Fluxions, with
their application to the most considerable problems

hi

XTl LIFE OF THE AUTHOR.

in Geomftry and Natural Philosophy. This wcvk
was published at Edinburgh in } 742, 2 vok. 4to. }
and as it cost him infinite pains, so it is the most
considerable of all his works, and will do bim im*
mortal honour, being indeed the most cmnplfte troa^
tise on that science that has yet appeardd.

In the mean time, he was continually obliging the
public with some observaticm or perferaiauce of bis
own, several of which were publi^ed in the 5th and
6th volumes of the Medical Essays at Edinburgh.
|l^any of them were likewise published in the Pbilo*
sophical Transactions; as the following: 1, On the
Construction and Measure of Curves, vol. 30. «^
S. A New Method of describing all Kinds of
Curves, vol. 30. — 3. On Equations with impossible
Roots, vol. 34. - v^- On the Roots of Equations, Ac
vol. 34. — 5. On the Description of Curve Lines^
vol. 39. — 6. Continuation of the same, vol. 39- "-^
7. Observations on a Solar Eclipse, vol. 40. — 8. A
Rule for finding the Meridional Parts of a Spheroid
with the same Exactness as in a Sphere, vol. 41.-—
9. An Account of the Treatise of Fluxions, vol. 42.
— 10. On the Bases of the Cells where the Bees
deposit their Honey, vol. 42.

In the midst of these studies he was always re^dy
to lend his assistance in contriving and promoting
any scheme which might contribute to the public
service. When the Earl of Morton went, in 1789,
to visit his estates in Orkney and Shetland, he re*
quested Mr. Maclaurin to assist him in settling the
geography of those countries, which is very errone*
ous in all our maps ; to examine their natural his^
tory, to survey the coasts, and to take the measure

of

Uf £ OF TBX AiJT&OR. Xiu

of a dcgi^e of the incridiaa. Maclauria s family
affairs would not permit him to comply vith tfau
MqvGSt : he irem up^ howew^ a snanorial of what he
thought tteca^aiy to be observed^ aodiiuoitriicd pro^
per iastrai:)eiti fhor the work, recommending Mr.
ShoTt, the not^ optidfiiii, as a £t operator £x the
mafio^eoMUt of them.

Mr. Maolaurin had still anotSner scheme for tfat

inprovement of geography and navigatioci^ of a a&ore

f!ic:tGii5ive nature ; which was the opening of a pas-^

lag^e from Qtednhaod to the Sooth Sea by the North

Pbte. That :Mtsh e passage might be found, he was

00 ftilly (p&fpmiBA, tisat ha used to say, if his atuat

tioa oo«td admit of tudh ad\'entureB, he ' would nn*

deiitaibe ihe vvpyage efven at his owin charge. But

nAen schemes fiixr Aiding it were laid before the

Parliame^ ln'1744, and he was consulted by several

pcopmas of high yank conoemmg them, and before

he omild Aiuah the memorials he proposed to send,

Iii6(premimn was lonitodl to the discovery of a North

Wear 7»ttage : and he used to regret that the word

West was inaerted, because he thought that passage,

if «t»sdlto be found, muat die not far from the Pole.

* in 1J745, having "been very active in fortifying the

oisty letf Sdmburgh against the rebel army, he was

efa%^ to ily £rom thence into England, where he

waa invitisd by Dr. Heniqg, Archbishop of York, to

reside wMi him during hb stay in this country. In

dna eKpedxtioa, howeter, being exposed to cold and

hardfeftiips, and isaturaUy of a ^^'^eak and tender con*

stittttum, wfakb bad been mnich more enfeebled by

close appliortion to stud}% he laid the foundation of

on ttntts niikjh put an end to 3iis lifi^ 'mJuae 1746,

at

Xvm riFE OF THE AUTROSi

at 4A years of age» leaving his widow with two udia
and three daughters.

Mr. Maclaorin was a very good as well as a very
great man, and worthy of love as well as admiration^
His peculiar merit as a philosopher was, that all his
studies were accommodated to general utility ; and we
find, in many places of his works^ an application
even of the most ahstnise theorite to the pcHrfecting^ •
of mechanical arts. For the same purpose, he had
resolved to compose a course of Practical Mathe^
matics^ and to rescue several useful bran:cl»s of the
science from the ill treatment they often met with in
less skilful hands. These intentions, however, were
prevented fay his death ; unless we nlay reckon, as a
part of his intended work, the translation of Dh
David Gregory's Practical Geometry, which he re*
vised, and published with additions, in 1745.

In his life-time, however, he had frequent oppor*
tunities of serving his friends and his country by his
great skill. Whatever difficulty occurred concerning
the constructing or perfecting of machines, the work«
ing of mines^ the improving of manufactures, the
conveying of water, or the execution of any pubUo
work, he was always ready to resolve it He was
employed to terminate scene disputes of consequence
that had arisen at Glasgow concerning the gauging
of vessels ; and for that purpose, presented to the
commissioners of the excise two elaborate memorials,
with their demonstrations, containing rules by which
the officers now act. He made also calculations
relating to the provision, now established by law, for
the children and widows of the Scotch clergy, and of
the professors in the Universities, entitling them to

certaiv

1

LiVE OF THE AUTHOR. xiz

certnin aniiuities and sums upon the volontary
annual payment of a certain sum by the incumbent.
Jn contriving and adjusting this wise and useful
scheme he bestowed a great deal of labour, and con-
tributed not a little towards bringing it to perfection.
Of his works, we have mentioned his Gtometria
Organica, in which he treats of the description of
curve lines by continued motion ; as also of his Piece
which gained the prize of the Royal Academy of
Sciences in 1724. In 1740, he likewise shared the
prize of the same Academy with the celebrated D.
Bernoulli and £uler, for resolving the problem delat-
ing to the motion of the tides from the theory of
gravity ; a question which had been given out the
former year without receiving any solution. He
bad only ten days to draw up this pap^ in, and could
not find leisure to transcribe a fair copy ; so that the
Paris edition of it is incorrect. He afterwards
reivised the whole, and inserted it in his Treatise of
^piuxions ; as he did also the substance of the former
Tiece. These, with the Treatise of Fluxions, and the
Pieces printed in the Medical £ssays and the Philo*
sophical Transactions, a list of which will be found
in page xvi, are all the writings which our Author
lived to publbh. Since his death, however, two
more volumes have appeared ; his Algebra, and his
Account of Sir Isaac Newton's Philosophical Dis-
emeries. The Algebra, though not finished by him-
self, is yet allowed to be excellent in its kind ; con-
taining, in no large volume, a complete elementary
treatise of that science, as far as it has hitherto beea
carried ; besides some neat analytical papers on curve
linos% His Account of Newton's Philosophy was occa-
sioned

kx tjfz OF THE AtyrnotLi

sioned in the following ifianner : — Sir Isaac dying in
the beginning of 1728, his tiq>hew, Mr- Conduitt,
proposed to publish an Account of his Life> and desired
Mr. Maclaurin's assistance. The latter, out of gra-
titude to his great benefactor, cheerfully undertook^
and soon finished, the History of the Progress which
Philosophy had made before Newton's time; and this
was the first draught of the work in hand, which,
not going forward on account of Mr. Conduitt's
death, was returned to Mr. Maclaurin. To this
he afterwards made great additions, and left; it
in the state in which y it now appears. His main
design seems to have been to explain only those parts
off Newton's Philosophy which have been contro*
verted : and this is supposed to be the reason why
his grand discoveries concerning light and colours,
arc but transiently and generally touched upon ; for
it is known, that whenever the experiments, on
which his doctrine of light and colours is foimded^
had been repeated with due care, this dodtrine h^
not heen contested ; while his accounting for the
celestial motions, and the other great appearasices of
nature, from gravity, had been misu&deFstood, and
cvep attempted to be ridiculed*

dOKtENTS,

^ %

«

XABtB: op. TH£ PKQICIPAL C(»?TQITS<

Ti

« •

INTRODUCnON,

£fiyi^«iHf armto*4rM€(N^ •«• 9

2^ giiadratiire <2f a iqpin^ 31

« t . p « • • •

» « « •

BOOKI.

to^ttikums'a$id tfltf^o&M^, ^ .Article I

The dsHomif .••'••'/.••• • • . 15

T%«ore»» concerning wijfbrm motions from Archimedes, ••••••••• 1$

Theorems concerning vmiable motums,** « It.

Ofcomfarh^ tliejhuaons rfquanivties hy deteifimmg the Undt of the

raHo of their incrfments or decrements^ 66

Qf second JkoBionSf • 70

CHAi^. H/ (Jf thejkaaans of ftoane rectilineai fgures.

Of the Jhixwn of a paraUetogram of an invariable aUitudef • • • •Art. n

QftheJhuAmrfatrianglef.. »»»..»....,. •••••••• ^l

c ««

The merimmi ef IRe triangle raolvedinio twoparte^'^hsi ti»Udl
$neatmt$ the generating motion^ and that whkk mauuree He

acceUrationf « ••••••* ^3

The theory of nuftions that are accelerated or retarded wn^hmdy^. • 94.

Cfi/iejhi9am.i^attctiingle^... 4.114 9^,

Chap. III. Of the Jiuxkmsrfplafiecurvilineal ^figures*

Of the^fiuxionqfanareat the ordinatei being stg^fOied parattelf Art 105

General doroUarieerdating to the theory (fmotianf 114

Ofthefwaion of the area generated by a rOy reooking tAoutagicen

centrCf II6

Ofdmilarcurvilmeal^figures^. ••• •• ••• 122

Chap. IV. CftheJiuxmsofaoUde,...., Art. 124

lUwtratians of second and third fluxions^ • ^^^^

Chap. V. Of thej/hmoneef- quanHtiea that are in a continued

geometricalprogreesimythefirsttermofwhkhieinfoariMe^ Art. 140
Chap. VL ^OfiogarithmSt ^ thejhueions qfU^arithmie quantities.
An accoimt &f logaritJms from Napier the inventor^. •*•«•» .Art 151
Ofthe^^naonsrfquantiiiesihatinereaseordecreaseprofOTd 158

Hftheflnnonsof quantitieSfSohen their logarithms arein ahvmoarir

able ratiOf.... l6#

€^theJlnaione4fqamtitksthat'are'representedhyp(r^

tionalor variable enponente^m - - •► i* ........* i * * « l6$

Of the sMomdftUrdf and highcrfluaitjnsif a quantity that increases >

propoitioiiaUy,..«....%.^. •••••••.«•••• I69

Theoremsforapprommating to the value rfiogmithmSf..^ 17I

Of the d^estn$ logarithmic systems and the rn^omodvils^^ 174

Of the logarithmic curve^ • 176

Of hyperbole areas 177

Of the analogy betwixt circular, arks and logarithms^ 17S

Chap. VII. Of tangents.

D^nitionsy • Art 180

OftheftusAon of the base, ordinate^ and curve^^ ••••^ 184

Ofthejluxum of the arif me^ tangent] secant^ &c ISi

Of the fluxions oftVt curdcy the ray drawn to the curve from a given

pointy and the circular ark described from that point as centre, • • 199

Cfthejluxions qfmgles^.W • . • • * 208

Theorems concerning tangents^ • •»•••• 211

ChapI Vni. Of the fluxions of curve surfaces.

Lemmas concerning caiucal surfaces'fi ••«•••••••••• .Art. 21$

()f the fluxion of a curve surface^ #••» •««,«#»••••###••••••#• • 22i

Of the nafaces generated by amf arkSf the eetttre rf grmty ef m

mxk, and the tktorem (f G uldmtu^ • 4 • • ; ; ; ; i 2S)

CsA?. IX. OftkeimialnitfQrditembmgtJ^grtattdmtdletut

crdhuUcSf ».,»,»» u%;ii.u.«;;*;;.;;Art; S9S

Oftk analoofhctuist. tAeiaverse method (f taMgwtr, mi the fioaK

dr9iureqfjigmt$f i ••• 847

AmoreiKcm-ater^farJmdmgtkegreatettmdleaticr^^ S6i

A nmiOar fMkfar^findmg tiiefoinU ffeoK^^ 9ff8

Of cmgUk of 'oaiwia kkuky... •••• * HM

OftM greatat andkattraif$ that can he dramafnim agkmpomt to

a <i«hp^« «»«•.•• •••«»••»•*»••• ••••WI44 fSTf

Other rtUeafir^fiadmg the pmU (if emtrmrffk^ 979

Chap. X. Cff thfi,aiii/iBiiptQte9 cfernve tma, &c.
T^efMiUmtfaggmftatee^v^ 28<

^thtfortiffgeomOxi^mfigtati^
Ofaifi/iptote$fimdtheareQ4b<madedhyihta^andtkecmnom^ii

Qfthe,^(didgenerated.hytU$ana,^»..^.^^^.^.^ «.44.» 507.

Examples ofeonttructiotiBfQrdettrmkmgtietaHgeats aadaUffmtMet

cfcwveethat Oredeeeribed bythereoobikmaf Uaeearaagkh 3iS
^I^^c^>r€mJar.4»$ca!oerin^ whether a Jigmrekatkm aefmftatOf dad
iha areahamded byit aadm^^m^ kaikaa aeeigaakk Uadi

t^ihc Mufaug^^atedhytheak^dbemithtatfaipto^

Qf^iirallmes aMiiMfrinvvEi>.*«*««4*^«*«»**««««^ ••••««•• 340
CftJke Umte to which the sums ^ progressums approach^with exam^

flee aad theorems for apfro s ma t 'mg to those limits^ «« 350

Chap. XL Gftheearvatare qfUnes^ &c«

D^Gmltoiw, 4«..«4.«Art. 368

theorems for fmdiag the curvature aad its variation m geometrical

JigureSf and for comparing the d^erent degrees of contact of the

euroe and drckrfcutoaturefk •••«•*••«•*».«••* i.*4« * •• * S6i
fxoffip&f tfi^Ae come MCiiofMy. 4* *•**••«*••*•*•*•«•• *••*«« 371
Cff the cwrvature that is less that that fifai^cirde^.**kk^..s*.* 377

Of the curoature that is greater than in any drckf* • • • * • • 378

Otieriheorems concerning the curoature and Us mruUkoj 381

A general property oftheHnes of the third orders when tvfo tangents

can be drawn to the Une from a point in it^ ••••••*«.4.» 401

(ifthewMonQfHnes^ «•«,#•«•#•#*.*..«>•« 402

• a Of.

CJ^Aerih^/iU<4s^.^,^Sri<Mi4icoiM^faJ^^ 415

centre or recedesjhmi^ ^^« •* ^ •«*«•*> i ••.#•««• «r «•>, •#«»•* • 44f
Ifftkt rtmimM tmd dt^ rfike medtm^^Ma n^fkm trmffe-

tmf, it. 4efCf*iM,. ..•.^^^•.•.« # » • • * , • ^ • ^ » ^ • • • * • m • 4 m • • ^M

efgravUatimtmm^ ememf^dei^km,^ ...,*... ./.. *• . 46*

^tkeM^Uon vf4ki ^mm^m ^tk^wwm,**. . .4 •/•*••••••••••• « ^480

<^#ife«MBl^0fM04i<M^^^ 48r

QTi^A* iu§aefmiimoffktmti9k fhte^thfig tJb MiMldM^ tib ettfik, ^ 490
QO^iMM|r|>i^ 491

tytk$inierwe€H0^^t»$ 9m^ ' m ^xi rio k ^4urHti9r6,4 4 • . « I vv'. ; '^sii

» , »

«w > • )'

* • • » « • 4

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(• •■•!••

t • • t

INTRO-

A*».

mTRODUCTlOK.

ItEOMETRY is valued for its extensive useftilness^ but has
been most admired for its evidence ; mathematical deihon-
stiation being such as has.been always soppoated to put dn end
to dispute^ leavingno place for doubt or caviL It acqattddthis
character bj the great care of the old wxiterB^ irho adbutted
no principles but a few self-evident tmths, and nOdemoiislnra«'
tions but sQch as were accurately dedticed from tinem. The
science being now vastly enlarged, and applitd vfiVk suocess to»
pbilosophj and the arts, it is of greater importaaee than ever
that its evidence be preserved perfect, fiat it has heed oU^
jected on several oceasicma, that the modem imtpHfOfmAeaAm
have been established for the most part upon new and cxc»fi«
tionable maxims, of too abstrose a natajie to deserve a placQ
amongst the plain principles of the ancient gefametry : alid
some have proceeded so far as to iittpate false reasonii^ to
those authors who have contributed most to the late diaoOveM
ries^ and have at tbe aame time been most cautious in their
manner of describing them*

In the method of indivisibles, lines Were ctaci^ed io .btf
made np of points, sorfaces of lines, and solids of surfbees; imd
such suppositions have been employed by several bgemoni mto
for proving the old theorems, and discovering new ones, m 4
brief and ea^ manner. But as this doctrine was inconsistenk
with the strict principles of geometry, so it soon appeared that
there waa some danger of its leading them into false conclu-
sions : therefore others, in the place of indivisible, substituted
ii^nttely small divisible ekments, of ^d!Kick they auppoaed all
magnitudes to be formed ; and thna endeavoured to retain, and
ilniHrove, the advantages that were derived from the former me-
thod for the advancement of geometry. ^After these came to be
relished, an infinite scale of infinites and infinitesimals (ascend-
ing and descending always by infinite steps) was imagined and
proposed to be received into geometiy, as of the greatest use for
penetrating mto its abstruse parts. Some have argued for quanr

VOL.1. A tiliea

^ tKTR0l>XrCTI07f4

titles more than infinite ; and others for a kind of quantities
that are said to be neither finite nor infinite^ but of an interme-
diate and indeterminate nature.

This way of considering what is called the sublime part of
geometry has so far prevailed^ that it is generally known by no
less a title than the Science, Arithmetic, or Geometry of infinites.
These terms imply something loffcy^ but mysterious ; the con-
templation of which may be suspected to amaze and perplex,
rather than satisfy or enlighten the understanding, in the prose-
cation of this science ; and while it seems greatly to elevate
geometry, may postibly lessen its true and real excellency, which
chiefly consists in its perspicuity and perfect evidence ; for we
may be apt to rest in an obscure and imperfect knowledge of so
abstruse a doctrine,as better suited to its nature, instead of seek-
mgfor that dear and full view we ought to have of geometri-
cal truth ; and to this we may ascribe the inclination which has
appeared of late for introducing mysteries into a science where-
in there ought to be none.

There were some, however, who disliked the making much
nse of infinites and infinitesimals in geometry. Of this num-
ber was Sir I$aac Newton (whose caution was almost as
disttngnishing apartof hischaracter as his invention)^ especiaUy.
after he w^ that this Uberty was growing to so great a height.
In demonstrating the grounds of the method of fluxion*, he
avoided them,e8tablishingitinawaymore agreeable to the strict-
ness of geometry. He considered magnitudes as generated by
a flux or motion, and showed how the velocities of the gene-
rating motions were to be compared together. There was no-
thing in this doctrine but what seemed to be natural and agree-
able to the antient geometry. But what he has given us on this
%ubjectbeing veryshort^hisconcisenessmay be supposed to have
givensome occasion to the objections which have been raised
against his method.

When the certainty of any part of geometry is brought into
question, the most effectual way to set the truth in a full light,

and

I

XNTRODUCJIOir. S

and to prevent disputes^ is to deduce it from'aiioms of first
principles ofunexceptionableevidence^by demonstrations of the
strictest kind^ after the manner of the antient geometricians.
This is our design in the following treatise ; wherein we do
not propose to alter Sir Isaac Newton^s notion of a fluxion^ bat
to explain and demonstrate hismethod,by deducingit at length
from a few self-evident truths^ in that strict manner : and, in
treating of it, to abstract from all principles and postulates that
may reqiure the imagining atiy other quantities but such as mi^
be easily conceived to have & real existence. We shall not
consider any part of space ol time as indivisible, or infinitely
little ; but we shall consider ^ojnt As a term or limit of a line;
and a moment as a term mc ttaii of time : nor sbaU we resolve
curve lines, or curvilineal spaoes^ into rectilineal elements of any
kind. In delivering the principles of thismethod. We apprehend
it is better to aVoid such suppositions : butafter these are demon-
strated, shortand concise waya of quaking, thoughless accurate^
may be peimitted, when ther^ is no hazard of our introducinj;
any uncertainty or obscoriiy into the science from the use of
them, or of involving Wiu. ^^^iites. The method of demonstra^
lion, which was invented by the author of ftu^Vonc^ is accurate
and elegant ; but we propose to begin with one that is some^
what different ; which, being less removed firom that of the an-*
tients, may make the transition to his method more easy to be^
ginners (fbr whom chiefly this treatise is intended), and may
obviate some objections that have been made to it.

But, before we proceed, it may be of use to consider the steps
by which the antients were able, in several instancies, from the'
mensuration of right-lined figures, tojudge of such as were bound-
ed by curve lines ; for as they did not allow themselves to re-
solve curvilineal figures into rectilineal elements, it is worth^
while to examine by what art they could make a transition from
the one to the other : and as they were at great pains to finish
their demonstrations in the most perfect manner, so by follow-
ing their example, as much as possible, in demonstrating a me-
thod so much more general than their's, we may best guar4
against exceptions and cavils, and vary less from the old foun-.
dations of geometry.

As They

Thejr lowid^tlisi libiilar triangles are to eadi cfUier in tbe dn^
pUcate ratio of dicir homologous sides ; and, by iesol?ing simi-
lar pdjygons into similar triangles^ the same proposition was ex^
indad to these potygons also* But when thej came to oom^
pane cumUneal figiirq^ that cannot be resolved into rectilineal
parts^ this method failed. Ordes are the only curviHneai plane
fgures considered in the ek&Khts t>f geometry. If they could
have allowed themsehes to have c«nsidered these as similar po-
fygoos of an iafimte ttvmbcfi of sides (as soh^ have done who
pretend to abridge their dononstrationsX after proving that any
similar polygcfns injicribed in circles aie in the duplicate ratio of
^e diametosy they would haveHlmmediately extended this to
Ae ohcks themselves ; and wbi^ have considered' the second
proposition of the twelfth bbok^ tl^eSlements as an easy co«*
itoQaiy firom the firM« But there* is grpund to think that.they^
would not have adnutted a demonstration of thiskind. It was a-'
^damental principle with them, thok the difference of any two
unequal quantities, by whichthe greater exceeds the lesser^may
he added to itself till it shall exceed any proposed finite quan-i
tily of the same kmd : and that th«f founded their prapositiona
aonoaming omvitt aeal figures upon this principle in a particular
manner^ is evident from the demoqstrafciDnsy Imd from the ex-
press declaratioa of Archimedes who acknowledges it to be
th^ foundation upon which he established his own discovoies*^
i^d cites it as assumed by the antients in demonstrating all
their propositions of this kind. But thia principle seems to be
inconsistent with the admitting of an infinitely little quantity or
^SkiSGoxx, which^ ijdded to itself any number of times^ is never
#uppos!^ to hficoiM equal to any finite quantity whatsoever. -

They proceeded th^acefoze in another manner^ less direct in-
deedj but p^];fectly evidsat* They found, that the inscribed si*
9iilar poIygo94^ by increasing the number of their sides, conti-
nually approached to the areas of the circles ; so that the decreas-
ing differeixoes betwixt each circle and its inscribed polygon,
by still further and further divisions of the circular arches which

t9ffiS^ Mrfvm Irl rS rfiyvMi, l?c* Archimcd* de quadr^parab. adDofith.

the

IlTTJtODVCTIOir* I

the sides of the polygons subtend^ could Become less than any
quantity that c^n be assigned : and tfaatdl this while the similar
polygons observed the saine constant inyariabte proportion to
each other^ viz. that of the sqniures of the diameters of the
circles. Upon this th^ founded a demonstration^ that the pro*
portion of the circles themsehres coald be no other than that
sam^ invariable ratio of thesimilar inscribed polygons ; of which
we shall give a brief abstract^ that it may appear in what maimer
they were able^ in this instance^ and some others of the same
nature, to form a demonstration of the proportions of carvilineal
iigures, from what they had aheady discovered of rectilineal
ones. And that the general reasoning by which they demon*
strated all their theorems of this kind may more easily appear^
we shall represent the cii*cles and polygons by right lines, in the
same manner as all magnitudes are expressed in the fifUl book
of the Elements.

Suppose the right lines AB t> -d r\ -d tx

and AD to repr^ent the two A Pj^B QEgD

areas of the circles that are *

compared together ; and let AP, AQ represent any ^mo simflat
polygons inscribed in these circles. By further eoQtinual subd»«
visions of the circular arches which the sides of the polygons
subtend, the areasof the polygons increase, andmay q>pioach to
the circles ABand AD so as to differ from them by less than any
assignable meastire; the triangle which is subducted ftom each
segment at every new subdivision being always greater than the
half of the segm<&nt. The polygons inscrH^ed in the two circles^
as they increase, are evar in the same constant proportion toeacb
other; and this invariable ratio of these polygons* must also be
the ratio of the circles themselves. For, if it ig not, let die ratio
of the polygons AP and AQ to each other be, in the ifst place^
the same as the ratio of thie circlie AB to any mugnitude AB
less than the circle AD; suppose the subdivisions of the archea
of the circle AD to be continued till the difference betwixt the
circle and insci'ifeed polygon become less than ED, so that the
polygon may be represented by Ay, greater than AE ; and let
Ap represent a polj^on inscribed in the circle AB, similar to the

A 3 polygon

6 INTRODUCTION.

polygon A;. ' Then, since AP is to AQ as ABis to AEby the
supposition, and the polygon hip is to the similar polygon A^ as
AP is to AQ; it follows^ that AB is to AE asApisto A^ ; and
that the cirx;le AB being greater than Ap, a polygon inscribed in
it^ AE must be greater than Ay. But A j is supposed to be greater
than AE \ and these being repugnant, it follows, that the po-
lygon AP is not to the polygon AQ as the circle AB is to
any magnitude (as AE) less than the circle AD. For the same
reason, AQ is not to AP as AD is to any magnitude (as AF) less
than AB. From which it follows, that we cannot suppose

AP to be to AQ as

F p B E 5 D t AB is to any mag-

^ — I — I ^ nitude Ac greater

than AD ; because
if we take AF to AB as AD is to Ae, AF will be less than AB,
and AP will be to AQ as AF less than AB to AD ; against what
has been demonstrated. It follows, therefore, that AP is not to
AQ as AB is to any magnitude greater or less than AD ; but
that the ratio of the circles AB and AD to each other, must be
the same as the invariable ratio of the similar polygons KS and
AQ inscribed in them, which is tl^e dupUc^te of the ratio of
their diameters.

In thesame manner the antlents have demonstrated, that py-
ramids of the same height are to each other as their bases; that
spheres are as the cubes of their diameters, and that a cone is
the third part of a cyUnder on the same base and of the same
height. In general, it appears from thisdemonstration,, that
when two variable quantities, AP and AQ, which always are
in an invariable ratio to each other, approach at the same time
to two determined quantities, AB and KS^y so that they may
differ less from them than by any assignable measure, the
ratio of these limits AB and AD must be the same as the in-
variable ratio of the quantities AP and AQ: and this may be
considered as the most simple and fundamental proposition
in this doctrine, by which we are enabled to compare cur-
vihAeal spaces in som^ of the more simple cases.

This

IKTRODUCTIOK. 7

This general principle may serve for demonstrating many
other propositions^besides the elementary theorems already men-
tioned. For example^ let ADB (p/.l ^^g.l) be ja semicircle de-
scribed on the diameter AB^ ABB a semiellipse described on the
saine right line as its transverse axis ; let AF6B be any polygon
described in the semicircle; and let FN^ GM^ perpendicular to
AB^ meet the semiellipse in H and K^ and the axis in N and M»
Because any ordinate of the circle is to the ordinate of the ellipse
on the same point of the axis as the transverse axis is to the con-
jugate^ it follows^ that the triangle ANF is to the triangle ANH,
the trapezium FNMG to the trapezium HNMK> the triangle
GMBto the triangle KMB^and the wholepolygon AFGB totbe

polygonAHKB^nthesameconstantratioofthe transverse to the
conjugate axis. Bisect the arch FG in D, and the triangle FDG
will be greater than half the segment in which it isinscribed. Let
the ordinate DI meet the ellipse in E^and the triangle HEK will
be also greater tha^ half the elliptic segment iq which it is in-
scribed ; for it is obvious that the right lines GF^ KH produced
meet the axis in the Mme point R^ and that the tangents at Dand
E meet it In the same point T ; con»c<^\iently the tangent DT
being parallel to the chord FG^ the tangent of the ellipse at E
IS pardlel to HK^ and the triangle HEK is greater than half the
segment HEK. The polygons^ therefore^ AFGB, AHKBj, by
continuaUy bisecting the circular arches, may approach to the
areas of the semicircle and semiellipse, so as to differ from them
by less than any assignable measure. Hence if the right line AD
in the preceding article represent the area of the semicircle^

AB the area of the semiellipse, AQ the ^ ^

polygon inscribed in the semicircle, AP A P B Q D
the corresponding polygon inscribed in
the semiellipse ; it will appear, in the same manner, that the
semicircle must be to the semiellipse in the same ratio that is
the constant proportion of those inscribed polygons, viz. that
of the transverse axis of the ellipse to the conjugate axis.

We have given this demonstration a little different from that
ofArchimedes in his fifth proposition of conoids and sphcroid^^
that the same proportion might ap^ar to be the ratio of the

A 4 wtf^

8 XKTRODUCTIOK*

area NFDGM in the circle to Uvs area NHEKM in the dJ6pe.
From which it follows^ that if C be the oommoa center of th«
cir-cle andellipse^the trianglesCFN/^HNand thetrianglesCGMt
CKM being to each other as tllfe tranverse axis is to the conju*
gate^the sectorCFDG in the cirdemijst be to the sectorCHEK in
theellipsein the same proportion. Let the diameter CE meetits
ordinate HKin P,and CP being to CE as CR is to CT,or as CQ
is to CD ; it appears^ that when the ratio of CP to the semidia^
meter of the ellipse CE is given^ then CQ is given> and there-
fore the sectorCFDGisofadeterminedmagnitude; consequent^
ly the elliptic sector CHEK^ and the segment HEK^ must each
be of a determined invariable magnitude in the same ellipse^ when
the absciss CP is in a given ratio to the semidiameter CE. The
triangle CHK> and the tri^>ezium CH^K (formed by the semi-'
diameters CH^ CH, and the tangents H^^ KO^ are also given in
magnitude when this ratio of CP to CEj or of CE to Ct, is given*

In general^ if uponany diameterproduced without the ellipse^
any number of points be takeny on the same or ondiflferent sides
pf the center^ At distances from it that are each in some given
ratio to that diameter ; and from these points tangents be drawn
to the ellipse in any one certain order; the polygon formed
by these tangents is always, of a given magnitude in a given
ellipse^ and is equal to a polygon described by a similar con-
struction about a circle^ the diameter of which is a mean pro-^
poitional betwixt the transverse and conjugate axis of the
ellipse. The polygon inscribed in the eUipse by joining the
points of contact^ and^ the sectors bounded by the semidia**
meters drawn to these points^ are also of given or determined
magnitudes ; and the parts of any tangent intercepted betwixt
the intersections of the other tangents with it^ or betwixt
these intersections and the point of contact, are always in the
^ame ratio to each other in the same figure. There is an
fLnalog04s property of Uie other conic sections.

When Archimedes demonstrated, that the area of a circle
is equal to a triangle upon a base equal to the circumference
pf the ci^cle^ of a height equal to the radius, it was not by sup^

posing

rUtelPa.S.Vt^^J

K
A

H

X

V

y^ 8

y^v

y^ B

F

o

X

I)S.T1L01>VCTI0S« ' O

ponng it tocomcide with a drcmmcribedequikteralpolygon of
aniofiQite rnunber of sides^butina more accurate and unexcep-
tionable manner. Let bd (fig. 2), the base of the right-angled
triangle^ cAd, be supposed equal to the circumference of the circle
ABD^ ah equal to the radius CA^ EFGH any equilateral poly-
gon described about the circle^ ABDK a similar polygon inscrib-
fsd in it^ and let CQ perpendicular to AB meet it in Q. As the
circumscribed polygon £F6H is greatei" than the circle^ so it
is greater than the triangle abd; because it is equal to a tri-
angle of a height equal to CA or ab, upon a base equal to the
perimeter £F6H^ which is always greater than bdxhtcixcamf
ference of the circle. The inscribed polygon is less than the
circle, and it is also less than the triangle abd, because ft b
equal to a triangle of a height equal to CQ (which is less than
CA or ab)f upon a base equal to its perimeter AB^K, wfaichis
less than the circumference of the circle bd. Ther^ore the circle
and the triangle abd axe both constantly limits betwixt the ex-
ternal and internal polygons EFGH/ABDK. Let the arch AB
be bisected in U «n4 th^ tangent at L meet AE, BE, in M and
N; and the angle ELM being ngka^BM mustbe greater than
JiMor AM^ the triangle ELM greater than ALM^EMN greater
thanthesumofthetrianglesALMjBLNyand consequentlygreat-
er than half the space EALB bounded by the tangents £A, EB,
and the arch ALB : firom which it fallows (by the 1.10. Eucl.
the foundation of this method), that the circumscribed polygon
may approach to the circle so a»to exceed it by a less quantity
than amy that can be assigned. The inscribed polygon may al-
so . i^tproach to the circle so as that their difference may become
leas than any assignable quantity, as is shown in the Elements,
Therefore the circle and the triangle abd, which are both li-
mits betwixt these polygons, must be equal to each othen For,
jf the triangle abd be not equal to the circle, it must either be
ipreater or less than it. If the triangle abd was greater than the
circle, then, since the extemalpolygon, by increasing the num-
ber of its sides, might be made to approach to the circle so as to
exceed it by a quantity less than any difference that can be sup-
posed to be between it and the triangle a&c^; itfollows,thatthe

lUit^nuJ polygon fli%bit bccoiM ksi than thattrian^^ against

what

10 rNTRODXrCTIOK.

what has been demonstrated. If the triangle a&d was less than
the circle, then the inscribed polygon, by being made to ap-
proach to the circle, might exceed that triangle : which, by
what we have shown, is al$o impossible.

In general, let any determined quantity AB be always a li-
mit betwixt two variable quantities, AP, AQ, which are suppos-
ed to approach continually to it and to each other, so that the
difference of either from it may become less than any assignable
quantity, or so that the ratio of AQ to AP may become less
than any assignable ratio of agreater magnitude to a lesser. Sup-
pose also any other deteimined quantity ai to be always a limit

betwixt the quantities ap
A P e B E Q and aq; and, aq being al-
ways equal to AQ, or less
than it, let a/i be either

m ' p b ' q equal to AP or great-
er than it; then shall
these limits AB and ah be equal to each other. For, itab was
equal to any quantity A£ greater than AB, then, by supposing
the ratio of AQ to AP to become less than that of AE to AB,
the ratio of ap to AP would become less than the ratio of ah
to AB: but since aq is always greater than ab, and APIess than
AB^ the ratio of aq to AP is always greater than the ratio of
4f6 to AB; and these being repugnant, it follows that ab is not
greater than AB. Ifab was equal to any quantity Ae less than
i\ B, then, by supposing the ratio of AQ to AP to become less
than that of AB to Ae, the ratio of AQ to ap (which, by the
supposition, is either equal to AP, or greater than it) would
-become less than that of AB to ab : but since AQ always ex-
ceeds AB, and ap is always les than ab, the ratio of AQ to ap
must be greater than that of AB to ab; and these being re*
pugnant, it follows that ab is not less than AB. These limitSE,
therefore, AB, ab are equal to each other.

I II this manner Archimed^i demonstrates the fourteenth pro-
position of the first book of his admirable treatise concerningthe
ssphere (ind cylinder. I^t Dhba (fig. 5) be an upright cone, D

the

IVTRODUCTIOK. , 11

the yertez^ C the center, and CA the radius of the base. Let
KI be a mean proportional betwixt DA^ the side of the cone, and
CA the radius of its base ; and the convex surface of the cone
shall be equal to a circle of the radius KI. Let LMNS be any
extemal*equilatera] polygon described about this circle, hnns a
similar polygon inscribed in it ; EFGH a similar polygon de-
scribed about the base of the cone; efgh a similar polygon de-
scribed in it : then the surface of the pyramid DEFGH, which
is described about the convex surface of the cone, shall be to
the polygon EFGH as DA is to CA, or as the square of KI ia
to the square of CA, or as the polygon LMNS is to the poly*
gon EFGH ; and therefore the surface of theextemal pyramid
DEFGH, without the base, is equal to the polygon LMNS.
Let CA, perpendicular to EF, meet efiuBj and BR, parallel
to DA, meet CD in R : then the polygon Inms being to efgk
as the square of KI is to the square of CA, that is, as DA is to
CA, or as BR is to CB ; and this being a less ratio than that
of DB to CB, which is the same as the ratio of Defgh, the sur-
face of the internal pyramid without its base, to the internal po-
lygon efgh ; it follows, that the surface of the internal pyramid
without its base is greater than the polygon Imm. The external
polygon LMNS is greater than the circle of the radius KI, the
internal polygon is less than that circle, and the ratio of the ex-
ternal polygon to the internal may become less than any assign-
able ratio of a greater quantity to a lesser. The surface of the
external pyramid DEFGH, without the base, is greater than
the convex surface of the cone ; and the surface of the internal
pyramid Jhfgh, without the base, is less than it. Therefore by
substituting, in the general demonstration of the last article, the
circle of the radius KI in place of the quantity AB, the exter-
nal polygon LMNS for AQ, the internal polygon bnm for AP,
the convex surface of the cone for ab, the surfaces of the exter-
nal and internal pyramids without their bases for aq and ap ; it
will appear, that the convex surface of the cone and the circle
of the radius KI are equal to each other. We have given so
full an account of this demonstration, therather;^ that a part of
it has been sometimes misrepresented.

It

m IVTRODVCTIOir.

It ismuch in thesamemanner thathedenionstrates the thhtjr*
Sr8l> fortieth^ and forty^first, or, according to the celebrated Dr*
BmrT<m^B ntimbers, the thirty-seventh, forty-ninth, and fifti^fth
of the same toeatiie ; which are esteemed by ao good a judge,
lor the uaefiikiess of the pn^sitions, the subtlety of the iaven-
tioB, and the elegance of the demonstration, amongst the most
Talnable disco^ries ingeometiy. In the first of these hedemon-
strates, that the surface of a ^here is equal to four'times the area
of a great circle of the ^here. Let AEBFDfbe (fig. 4^ be aa
equilateral polygon inscribed in t)^e generating circle of a
number of sides that is any multiple of the number four ; let
OHKhiitkh be a similar polygon described about the circle,
kaTing its sides parallel to those of the internal polygon ; and
let Be, Bb, Vfmeet the diameter AD in Q, C, and N. Suppose
the s^iiekcle ABD, with die inscribed and circumscribed poIy«
gons, to retolve on the diameter AD ; and the conical surface
genemied by the diordAE shallbeequd toa cirdeof aradius
that is a mean pr^^orlbnal betwixt A£ and EQ ; the surface
generated by the chord EBshaB be equal to a circle of a radius
tiM^isamean prdpovtknM betwixt EB (orAE) andthesun^
of Che perpendidulatfs £Q and BC ; and the surface generated by
:Aie whole perimelej of the internal polygon equal to a circle of
a radiaslhaA is a natean proportional betwixt A£ and the sum of
the peipeiidiciihi^ £e, Bd, and Tf. But supposii^ Be and Ffr
fo meet tifie diameter AD in R and S, the triai^les DAE,
AEQ, QeR, R6C, CAS, SFN, and DN/'are simUar ; and DE is
to AE, EQ to AQ, eQ to QR, CB to CR, AC to CS, FN to
SN, and/N to DN, in l^e mme ratio : and therefore the sum
c^ the perpeildicu>1ar8 &, BA, ¥fiato the diameter AD as D£
is to AE. From which it follows, that the surface generated by
the revolution of the internal p<%gon AEBFD is equal to a circle
of a radius that is a mean proportional betwixt AD, iht diame*
ter of the generating circle, and DE the right line drawn from
one extremity of that diameter to die anj^le E that is next ad-
joining to its other extremity, in the same manner, the surface
generated by the perimeter of the external polygon GHKLM
is equal to a circle of a radius that is a mean proportional be- ,
twixt MH (which is equal to OXor AD)> and GM the diameter

X of

of tlie ciicle described aboat diat polygon. Because DE is lest
than AD, but GM greater than AD, it folkyws that the surface
generated by the perimeter of the internal polygon is alvrays
kcs, and the surface generate by the perimeter of the external
polygon is always greater Uian a circle of the radins AD, which
(because AD is double of AC) is equal to four times the area
of the generating circle. The surface of the sphere is also itself
always a limit betwixt these inscribed and circumscribed sur-
&ces; and because the ratio of these surfaces to each other is
thednplicate of the ratio of GH to AE, or of CO to CA, it ap-
peara^ that by increasing the number of the sides of the poly-
gons the ratio of these surfaces may approadi nearer to that of
aqoality than any assignable ratio of inequaKty; therefore iy
substituting, ip the general proposition demonstrated above, the
aurfiace of the sphere m place of AB, the quadruple area of a
great cirde in^'place of ab, the surfaces generated by the cir-
^uBascrihed and inscribed polygons for AQ and AF; andsuppos-
iiig 419 and ap respectively equal to AQ and AP ; it will be
evident tbat the suxface of the sphere fu^ the quadruple area
of its generating circle are etpoA ibo <i94M other.

The surface generated by the perimeter ABBF, inscribed in
amy arch AF, ia equal to a circle of a radios that is a mean pro-
porti<»iaI betwixt DE and AN, which is less than AF, the mean
proportional betwixt AD and AN: and the surface generated
tgr the perimeter 6HKL, being for the same reason equal to
a circle of anMlios tl^ is a mean proportional betwixt AD and
GNj which exceeds the chord AF; it follows, that a circle of
ifte radiua AF is always a Umit betwixt the surfaces generated
by tlie perimeters of the internal and external polygons AEBF
aad GHKL* But the portion of the spherical surface generated
by the revolution of the arch AP is always a Umit betwixt the
same circumscribed and inscribed surfaces : and these being to
each oth^ in the duplicate ratio of GH to AE, or of C6 to
CA, and thereibie in a proportion that may approach to the ra-
tio of equality nearer than any ratio of inequality; it follows,
that the' portion of the spherical surface generated by the revo-
lution of the Bfch AF is equal to the area of a circle described

with

14 IKTHOD|UCTIOV,

with a radius equal to the right line AF. The surfaced^ th^e*-^
fore^ generated by arches terminated at A are as the squares of
their chords^ or as their versed sines ; and parallel planes^ which
divide the diameter of a sphere in equal parts^ divide the sur-
face of the sphere into equal parts at the same time.

From these propositions^ it is easy to see how he was able to
compare the sphere itself with the circumscribed cylinder or in-
scribed cone. Let CP, perpendicular from the center on AE,
the side of the polygon^ meet it in P; then the solid generated
by the revolution of the triangle CAE being equal to a cone of
the height CA upon a base equal to the circle generated by the
perpendicular EQ, and this circle being to the surfiBM^ generat-
ed by the revolution of the chord A E as the squaie of EQ is
to the rectangle contained under AE and EQ, that is, as EQ
is to AE, or CP to CA, it follows (such cones being equal as.
have their heights and bases reciprocally proportional), that the
solid generated by the revolution of the triangle CAE is equal
to a cone upon a base equal to the surface generated by AE of a
height equal to CP, LetBEproduced meet the axis in T; and,
for the like reason, the solids generated by the triangles GET
CBT shall be equal to cones upon bases respectively equal to the
surfaces generated by the revolution of the right lines ET and
BT, of the height CZ equal to CP: so that their difference (or
the solid generated by the triangle CEB)must be equal to a cone
of the height CP upon a base equal to the surface generated by
the right line EB. In like nianner, the solid >generated by the
revolution of the whole inscribed polygon is equal to a cone of
the same height CP upon a base equal to the surface generated
by the perimeter of the polygon, which we have shown to be
always less than four times the area of a great circle of the
sphere. For the same reason, the solid generated by the revo-
lution of the external polygon GHKLM is equal to a cone of
the height CO or CA, upon a base equal to the surface generate
cd by the perimeter of that polygon, which, by what has been
demonstrated, is greater than four times the area of a great
circle of the sphere. A cone, therefore, upon a base equal to
four times the area of a greatcircle of the sphere, of an altitude

equal

IlTTBODtrCTIOV. iS

%qual to th^ radius^ m always a limit betwixt the solids generate
ed bj the external and internal polygons. The sphere itself
is abo a limit betwixt these solids ; and their proportion being
the triplicate of that of GH to A£, or of C6 to CA, which
may approach nearer to a ratio of equality than any assignable
ratio of inequality ; it follows^ that the sphere is equal to such
a cone^ or to four times a cone that has its base equal to the ge-
nerating circle^ and its height equal to the radius ; and that the
sphere is to the circumscribed cylinder as two is to three. The
sector of the sphere generated by the revolution of the sector of
the circle CA£ is shown^ in like manner, to be equal to a cone
upon a base equal to the portion of the spherical surface generate
ed by the arch AE (or the circle described with the radius
AE), of a height equal to CA the radius of the sphere. These
are the thirty-second and forty-second propositions of the same
excellent treatise.

If we suppose the semicircle and semiellipse to revolve on their
commonaxia AB(/ig. 1 J^and togenerate a sphere and spheroid;
then the conegenex^tedby OaPDtvvm^ AFNshallbe to the cone
generated by the triangle AHN in the dupiMsaU:^ xatio of FN
to HN> or of the axis AB to its conjugate ; because cones of
the flame height are as their bases. And in the same manner it
appears, that the soUd generated by the revolution of any po-
lygon AFOB^ inscribed in the circle^ is to the solid generated
by the levofaition of the correspondingpolygon AHKB, inscrib-
ed in the ellipse, as the square of the axis AB is to the square
of its conjugate. The solid generated by the tri^gle TID is
to the solid generated by the triangle TIE as the square of ID
to the square of IE, or as the square of the axis AB to the square
of its conjugate : and if any polygon be described about the
circle, and a polygon be also described about the ellipse, so that
the points of contact in the circle and ellipse be always in the
same perpendiculars to the axis,, the solids generated by these.
polygons shall be to each other in the same proportion. By sup-
posii^the circumscribed and inscribed polygons in the circle to
be equilateral, and to have their corresponding sides parallel, it
may be demonstrated; that the sphere b to the spheroid in the

dupli-

1$ iVTllODVCTtOir.

duplicate ratio of the airis AB to its conjngaiej and that the mm
Hds generated by the spaces VHMQ, HNMK aie in the same
proportion. The spheroid^ therefore^ is quadruple of the cone
of a height equal to hcdf the axis AB, on a base equal to the
circle described upon the conjugate &xis as its diameter.

As the segment of the ellipse HEK^ and the triangle HEK^
are of a determined invariable magnitude in a given eDipae^
when the ratio of CP to C£ is given ; so the portion of the sphe-
roid (generated by the ellipse revolving on its axis AB) which
is cut off by a plane through HK perpCT<ficular to the plane
AEB^ ia of a determined magnitude^ when the ratio of CP to
CE is given. A cone upon the same base with this portion^ that
has it& vertex in E^ or in t, is also of an invairiUUe magnitude,
when that ratio is given ; and thejre is a geMral property of the
circumscribed and inscribed solids analogous tothatabove*men^
tioned of the circumscribed and inscribed polygons.

Archimedes takes a different way for comparmg the Bpbit*
roid with the cone and cyluMter, that is more general, and has
a nearer analogy to the modem methods. He mtpposes the
terms of a progression to inofttase constantly by the same differ*
ence, and dononstrates several properties of such a progiession
relating to the sum of the terms and the sum of their squares $
by which he is able to compare the parabolic oonoid, the spie*
roid and hyperbolic conoid^ with the oone^ and the area of his
spiral line with the area of the circle. There is an analogy
betwixt what he has shown of these piDgfessions^ ami the pro*
portions of figures demonsttatad m llie e leme n tar y geometiy ;
&e consideration of which may ilki^&^te his dcxrtDBe^ and ae^
perhaps to show that it is'mOfe re gator aiid«om|dete in itakind
dian some have imagined*'. The leiaftian of the sum of the
terms to the quantity that arises by taking tibe gi e ate afc <tf them
as often as there toe terms> i» ifluslrated by coaqpavi^ the tii«
angle with the parattetegrua of the some beight and baae ; and
what behas^demonstrafeedof thestdsaof the aqmesof the teoM

N com-*

Xi^TRODITCTIOlf. tj

.. - . '

^ompiaiied with the square of the greatest term, maybe illustrat-
ed hy the proportion of the pyramid to the prism^ or of the
eone to the cylinder, their bases and heights being equal ; and
by the ratios of certain frustums or portions of these solids^which
^ay be deduced from the elementary propositions.

The base ABof the rectangle ABCD (/fg. 5^ being divided into
Any nnmberof equal parts, AE, EF, FG, GB, and the perpendicu-
fere ET, TV, 6R, being drawn meetingAC in H, I, L, and CD
in T, V, and R; the rectangles AH, EI, FL, GC may represent
a progression of terms that constantly increase by a common dif-
foence equal to the least term AH. The sum of these rectangles
form the figure AKHMIPLRCB, which is described about the
friangleACB, and is always greater than the triangle: their sum
without the greatest term GCis equal to the figure EHNIQLSB,
which is inscribed in the triangle, and therefore is always less
than it. The rectangle ABCD is the sum of as many terms
equal to the greatest GC,asthereare terms. And as thetriangle
ACB^ or one half of the rectangle ABCD, is always a limit be^
twixt these circumscribed andbi«cribed figures ; so, in general,
i£ the greatest term of any progression of thiskmd he taken as
often as there are terms, then one half of the quantity tl^at ii
thus produced shall be always a limit betwixt the sum of all the
terms^ and this sum without the greatest term; which coincides
ifith the first proposition of the treatise of conoids and spheroids.

Suppose the same figure to revolve upon the axis AB, and
while the triangle ABC generates a cone equal to the third part
of the cylinder described by the rectangle ABCD, the rectangles
AH, EI, FL, GC generate cylinders of equal heights ; which
are therefore as the squares of the right lines EH, FI, GL, BC,
that increase constantly by the same difierence equal to the first
tierm EH. The sum of these cylinders, or the solid generated
bythefigiure AKHMIPLRCB circumscribed about the triangle
,ABC, always exceeds the cone generated by that triangle :
Their sum without the cylinder generated 'by- the rectangle
GC, being equal to the solid generated by the inscribed figure
EHNIQLSB,i8 always less than the same Cone. The cylinder
'V<k..l B generated

iq XSTTAODUCTIOV*

g^crafted by tlie lectangle ABCD is the sum of as many (yji^m
deis^gaal to that geoerated by GC, as there are terms. ^4
fui the cone generated by the triangle ABC^ or osie third part of
the cylinder generated by ABCD^ is always a limit b^w?xt \hm
solidsgenerated by the figures AKUMIPUiCB,£HNIQf^6i
so^ in general^ one thirdpart of the quantity that is produced by
^ling the square of the greatest term of any prqgir^s^op Qfl|iis
kind as often ^ theFe are term^ is alwftys a limi{^ betwixt (ba
fvns of the squqaes of sjlH the termSj, and the saqie sip| yi!Pt4iQ|]|t
the square of the greatest t^nu. Archimedes dqfio^^^t^.
this proposition in a general way^ and by it tb^ pfRpoi^tkm 9i
the area of his first qpiral line to &e circumscribed cirde. But
pappus^ makes use of the ratio of the cone to the cylind^ for
the same purpose : and since we have the example of sq accurate
an authorjr we sht^l proceed to describe ArJUmediU*^ Q\hw
Iheojreins of thisk ind in the same maimer.

The eleTenthpcopositionQf the treatise conoeming spiralliQea
§ppeaxs£ram comparicfgibe cylindergenerated by theractangl^K.
EBCTwith the firustom of a cone generated by the tra^ppzium

£BCH,andwithth«3soIidsgeneratedbythe^g^r^£MIPLRCBA
EHNIQLSB> the former of which is always grea^r than th^fc
fnistumi and the latter less than it. For the cylinder generat-
ed by the recti^gle £BCT is to the iT^st^lm of a cone g^ae^
ratedby the trapezium EBCH^ as the square of BC is to tb^
rectangle contained under the right lines BC and £H added to
one third part of the square of the difference of these lines. Ttie
demonstration of which we shall subjoin^th^ratherthikt thisp]:09>
position will be of use afterwards.

Tfaerestoftheconstractionremuningasinthe5thfigiii6 (figS,
pL £), let £S parallel to AC.andHZparallel toA9^meetSCio$
andZ; and^ supposing BX to be a mean pioportipni^ betwixt
BC and EH^ or BZ^ complete the paraUelogram EBXY., Thm
let the rectangles EBCT, EBXY, by revohrmg aa the a»^ ABx
generate the cylinders TCct, YXxy, and the triangles AB&

* Colkct MaUicni, lib. 4. piop. SI.

AUtI, AET, EdC, EBS, EdX, BBZ geaerate tUe cones ACc^
KHh,ATt, ECV;,ES/,EX^, fiZz^ ami thesdiidgenei-aled bjth^
' psffallelo^ani ESCH shefl be equd to the cylinder YXxy i
as may b^ ^own from thb knovni proposition^ that the cabe
Of BC exceeds the two cubes of BZ aiMl' CZ (or BS) by thre4
times the parellelepipedon upotk the Mjuare of BX of the heighi
(7Z : iTOni which it f(^w9^ thai the cone ACc exceeds th^
tones ARA^ ESlby a solid ^ilal to tiie cylinder YXxy. Thi
same theorem ma.y beddoioAstrated firdm Uie tSbeactietitBiij propo-
sitions thus : the solid generated hj the parallelogram ESCit
18 the sum of these which axe generated by the triangles ESC
mad ECH. Tfart wkic^ is gkmm^ hy tSC is equal to the
exc€!Bs of twice the cone £X^ aborve the cone EZa, because the
mjtiaM of ^ <7- fi- £ttclO add«d to the sqtilsr^ ^BZis^qual
far two rectlkBgles under BC and BK (or twic<^ llie square' 6t
tX) added io tke ^uaie of C2 Or BS; ted ti^#efite^ tfie dif-^
fer^ice ofthe cones ECc, iSi (or the soUd gei^^ied by ESC^
is equal to Iheeiieess of twice the co&d ^BXifklkke the* con^
EZa. The solid geiuMrat«d by the triaa»|^ EdH is «qual td
flke exces* of tfcd scM g^iserfeted Vy ARC <or ttie cone ATt)
aJ^cyve the cone AHA. Bdt the sum of the squims of BC and
BX is to the sum of the squares of SX and B2 ib Bd is t6 JBZ^
ct as ABis'to A£; and therefore (l7^ 5. Eiicli^the di^rence

«f the squares of BC and BZ is ^ tS^e sum of the squares <^ BX
hjtd K; as EB is to AE: fiom whieh it follows that the <ih&

ib«nce of the eones Al!i,AMk/s^ the sotid gen^^ttedby ECH),

k equal to ihe sum of the cofiie* EX^ and BSSe. Therefore the

mHa generated by the paralldogram £SCH ii equal t6 thrice

Ae eohi EXx, or to th^ cj^linder XYj^; and the frustum

tSXkH is ec(us3 to thi^ cjlind^ adde^ l&'lSb^ eck€^ESs : so thtiC

fte eyfinder TCef is to the frustum IfCsA as ihb iiquai^ of Bd

is to the square of I^X (or the rectangle under BC and EH)

addeid^to one third part of the iit^isaf^ of BS^^ ^ CZ> tfaet(i£!te«

«ice of BC and EH. It app^, ilierdoie> that th6 cylii^deif

generated by the rectangle EBOTis td the solid ^endmted by

die figme BMfPLRCB in a lless pr<^ftloil^ and to thcf solid ge^

aerated by flie figure EHNIQLSS iii-a greater p!^oportion than

llie raHo of the square of BC 16 tjte i^ctangle under the right

Bfi Jines

Jt IKTRODUCTIOVf

lines BC and EH added to one. third part of ttie square of th«[
difference of these right lines. In general^ the quantity that
is produced by taking the square of the greatest term (in such a
progression as we described) one time less than the number of
^rms^ is to the sum of tiie squares of the terms without the square
of the least of them in a less proportion^ and to the sum of
their squares without the square of the greatest in a higher,
proportion, than the ratio of the square of the greatest term to^
the rectangle under the extreme terms added to the third part
of the square of their difference^

In like manner^ the third proposition Of the treatise concern*
ing conoids aad_ spheroids jq>peaira by comparing the solid gene->
rated by the r^ctaiigle HZCT with those generated by the tri-.
ingle HZCiind thf eg^res HMIPLR.CZ,NIQI^Z,reyolyingJ
abflut the axis^^i supposing EB tp be divided in^o any equa^
parts EF, fQ^ G3.. For the solid generated by wy rectas^le,!
NM beipg ^ the. solid generated by the rectangle NL as the
(iifference of %ls( ;sqtiares of FI and EH i» the difference of the
squares of Qhif^d £H> cm: as the rectangle containedunder.
NI and the aum of 2EH and NI (€. 2. Eucl.) is to the rectangle,
under WL and tlie $um of £EH and Wh, the solids* generate
ed by the rect^iig]fes NM^ WP, ZR increase in the same pro^s
portion as the q»ace$ described in that proposition, which are
supposed to be applied on a given line, (as 2£H)^ and to excee<^
by squares that have their sides ($s.Nl^ WL^ ZC) constantly^
increasing by a difference, equal to the ^ide of the first excess*
l^t the solid gcyierated by the triangle HZC is equal to the,
fQlid g^per^ted by the rectangle HX added to the cone ESs, bj
irhat .ha^ been deponistrated ; and therefore die solid generated
1^ the reptaogleilZQT ^ ^ ^^ sot^d generated by the triangle
^ZC as the difference of the squares of BC and BZ (or the rec-^
iangle CZ4;)iB to the dii&rence.ottjbe squares of BXand BZ
iot the rQctangle CZB) added to one third part of the square,
of: BS, or CZ ; that is as 2^« or liie sum of BC and BZ is to
die sum of BZ and one, third part of CZ ; and therefore as the-
sum of 2EH and CZ is to the sum of EH and one-third part
p( CZ. Fi<>i9 wbi^ it foi^ivs^ that the soGd geAerat^d by

the tectfiingk HC is to that which is generated by the figure
UMIPLRCZ in a less proportion^ and to the solid generated by
NIQI^Z in agreater proportion, than that ratio. From this and
the preceding theorems Archimedes deduces many proposi-
tions nelating to conoids and spheroids, and various portions of
4he8e8olids ; themost considerable of which may bedemonstrat^
ed in the following manner, that differs little from his own me^
thod^andmay possibly be thesameby which hediseeveredthem.

Let AIC (fig, 7>) be a parabola, A the vertex, AB a part of the
ajds, BC an ordinate; complete the rectangle ABCD, and the
conoid generated by the parabolic area AICBaboUt the axis AB
shall be equal to one half of the cylinder generated by the rec-
tangle ABCD. Bycontinuallybisectingtheparts of theaxis AB,
let it be divided into the equal parts AE, EF, FO, 6B ; and let the
ordinates EH, FI, G Lmeet the curvein H, I, L, theright line DC
m T, V, R> and the right line AC in X, Y, and Z. Complete
the rectangles AH, FH, EI, GI, FL, GS, as also the rectangles
AX, EY, FZ, GO, En, F^, and Gs ; and the cylinders generated
by AH, EI, FL, GC, having equal altitudes^ we as their bases, or
as the squares of the ordinates EH, FI, GL, BC, and therefore
as the parts of the axis AE, AF, AG, AB, or as the rectangles
AX, EY, FZ, GC. From which it follows, that the cylinder
generated by the rectangle ABCD, the solids generated by the
circumscribed figure AKHMIPLRCB, and the inscribedfigure
EHNIQLSB, are in the same proportion to each other as the
rectangleABCD and the figures AAXmYpZRCB,EXiiY;Z5B;
and consequently one half of the cylinder generated by ABCD
is always a limit betwixt the solid circumscribed about the co-
noid, and that which is described in it. The conoid itself is al-
so always a Umit betwixt the same circumscribed and inscribed
solids ; and the difference of these solids being equal to the cy-
linder generated by the rectangle GC, which, by continually
Msecting the parts of the axis, may become less dito any solid
that can be assigned : it follows, that the conoid generated by
the paraboUc area AICB is equal to half the cylinder generated
by the rectangle ABCD. If a parabolic conoid be cut by planes
paraUel to each other, but oblique to the axis^ the sections are
1 B 3 similar

Ifi iNTao9VCtt0|r*

yixnikr elapses ; aadi^JPfir^i^of tha^Qlid cvtoffbj^ob^
plane is ahowDji in like^manneiot to l^ ecpial to one. b«}f of a cyw
linder of the same base and altitude.

]^tCA^CBO^.$>jbe the semiaxifi of an ellipse AIB; com*
plete therect^gte CADBi^^4i<>iii CIX I^tCAbedividedjbya
continual bisectionxii^totbf equal paitsAE^EF^FGjGC; diW
t^e or4^ate^^Ejilj, FI^ Qfi meetiag BD in T, V^R, mi CD m
X, Y, Z ; complete the rectangles AH, FH, EI, GI, FL, CL,
apd the xect^^Qgles DX, VX, TY, RY, VZ, BZ. Then, be-
cause the square of any ordinate EH is to the diiference of the
f qnares of CA and CEas the square of CB is to the squareof CA^
iuud the difference of the i^uaves of AD (or ET) andEX isto
the dilSerencje of the squc^es of CA a^d CE in the fame propor*
tion ; it £q11ow% that the sqi^a^ of EH is equal to the diftareooi
^f the squares, of ET and EXj and that the circle described bgr
^H is eq;aal. tp the difference of the circles described by BT
fndEX: so tjb^thecyUo^dergeiieratedby the rectangle AEHK
inust be eqnaltpthe solid generated btjr the rectangle STXil.
In like manner^ the solid gei^acated by Uie circumscribed figure
AKHMIPLRBC k equal to that which is gweiated by the
^gure DAXfnXpZGCB;,. which always estoeeds the sdid diat la
generated by the trjuaugle CDB ; and thesolid generated by the
inscribed figure EHNIQI^C is equal to that which is generated
by the figure T^ftY^ZsB^ which is always less than the solid
generated by that triangle* Therefore the dolid generated by
the triangle CDB and the part of the spheroid which is genera^
ted.by the elliptic area AIBC are constaotly limits batwizt the
9ame circumscribed and inscribed 8olid3 » and dke diffsrencie ol
these solids, bein^ equal to the cylinder generated by the reo*
tangle BGjiyhichjby continually bisecting the parts of the axis^
may become less than any solid that can be assigned ; it appears^
that the solids gei^ated by the elliptic area AIBCnsnd the tii*
angle DCB are equal, and that the portion of the qfdieroid gtte-
rated by the elliptic area A EH is equal to the solid generated by
the tiriangle DTX revolving about the axis AC. But the sc^ ge-
nerated by the triangle DX/r, by what has been demonstrated, is
to the solid generated by the sectangle DTXk (or the cyllndar
. » -generated

fdtentt^ bjr ABIIli^) as tbe ifam of EX and one Aird part o^
TXkfoibe sam efflEXand TX^ or as Ae sum of CA and
4Cfiistothe triplesnmofCAttidCB; and therefbre the por-
tion of the spheroid generated by the elliptie area AEH is to
like inscribed cone generated by theteiangle AEHas the stun of
flCA and CEisto the sam <^ CA and CE.

In the same manner^ CA (fig. g) being aby aemidiameter of
lEhe generating ellipse^ Bfr the conjugate diameter^Dd a tangent!
lift A, BD and kd parallel to CA^ suppose a cylinder BD^ft to hi
described about half the spheroid ABftn touching it in the cir-
tfamference of thesection BAin perpendicular to the jhoieABb,
iSnA suppose the cone CDrdu to be inscribed in this cylinder,
iThen^ if any plane cut HAb perpendicularly in the ordinate HA
(Aftt meets BB, bd, CD, cd mT,t,X and x)^ the sections of thS
spheroid, cylinder, and cone made by this plane being similar el-
lipses, and the square of EH being equal to the excess of the
square of ET above the square of EX ; it fcJtows that the el-
Bpse which is the section of flbe spheroid wilih that plane, is
^ual to the difference of the effipses that axe the sections of the
Cylinder and cone made by the sameplane. From which it may
fee demonstrated, as in the preceding* article, by continually
tjsecting the parts of CA, and by circumscribing and inscribing
e;^Mciders of equal heights about &e spheroid and cone, that
the portion of the spheroid AHA is equal to the excess of the
frustum of the cylinder IfTtd above the frustum of the cone
jyXxd ; and that the portion of the spheroid AHA is to die in-
scribed cone AHA of the same base and altitude as the sum of
iCA and CE is to the sum of CA and CE.

Let AIB (fig. 10^ be an hyperbola, O the center, AC a part of
Dieaads, Athevertex,ADatangentmeetingtheasymptoteOD
in ]>, CB an ordinate meeting OD in e and Dd parallel to the axis
kkd; let e/^ B6 parallel to the axis meet AD iny and A, and the
iPest of the construction be similar to that of the eighth figure.
The square of any ordinate EH being equal to the difference
of the squares of EX and ET, the circle described by EH is
equal to tfie difierence of the circles described by EX and ET,

B4 the

M IlfTEOBUCTIOir.

I

the cylinder generated by the rectangle AEHK eqaal ta the
solid generated by the rectangle DTXA ; and^ in like manner,
the solids generated by the whole circumscribed and inscribed-
figures AKHMIPLRBC^EHNIQLSCarerespectivelyequal to
thq solids generated by the figures DAXi»Yp2^£{2^ TXaY^ZkZ,
JProm which it follows^ that the solid generated by the triangle
J)de is always a limit betwixt the solid generated by the figoie
described about the hyperbolic area> and that which is general-
ed by the figure described in it. The portion of the conoid
generated by that area is always a limit betwixt the same solids :
and the difference of these soUds beingequal to the cylinder ge-
nerated by the rectangle GB^ which^ by continuing to bisect^
the parts of the axis^ may become less than any soUd that can
he assigned ; it appears that the portion of the conoid generated
t>y the area ABC is equal to the solid generated by the triangle
Dde. But this solid^ by what has been demonstrated, is to the
solid generated by the rectangle Dclef, or cylinder generated by
the rectangle Cb, as the sum of eAD and Ce is to the triple
sum of AD and C^; and therefore the conoid generated by the
hyperbolic area ABC is to the inscribed cone generated by the
triangle ABC as the sum of 2AD and Ce is to the sum of ADt
and Ce, or as the sum of 20A and OC is to the sum of OA
and OC. Any portion of the conoid cut off by a plane oblique
to the axis, is to the inscribed cone in a like proportion ; as may.
b^ demonstrated much in the same manner.

The following general theorem comprehends the preceding
propositions, and several others of the same kind. It contains ai
property which extends to all the solids that can be generated
by ally conic section revolving upon its axis, including the
sphere and cone, and may be of use in mensuration. A frustum
or portion of any such solid terminated by any two parallel
planes, and a cylinder of the same height with the frustum upon
a base equal to the section of the solid made by a parallel plane
that bisects the altitude of the frustum, differ from each other
always by the same magnitude, in the same or in similar sotids,
when the inclination of the planes to the axis and the altitude
of the fiiistqm are given. In the parabolic conoid this differenci^

vanisheiij

r

maiAeB, the frostutn bong always equal ia a cylinder of th«
same height upon the aection of the conoid that bisects tlie alti-
tude of the frustomj tfDd is parallel to its bases. In the sphere, the
£tistiim is always less than the cylinder by one fourth part of »
right angled cone of thesameheightwith the fnistum^or by one
half of asphere of adiameter equal to thatheight : and this dif<*
ference is always the same in all spheres whatsoever, when the
altitude ofthe frustum is given. In the cone, the frustum alwayt
exceeds the cylinder by one fourth part of the content of asi^f
YXviBx cone that has the same height with the frustum. In the
hyperbolic conoid, this excess is the same as in the cone gene«
mted bj the triangle OCe (fig. 10) formed by the axis OC, the
mBympUite Oc, and the perpendicular Ce, the altitude of the irus^
tams^and theindinationof the axis to their basesbeing the same
in both. In the spheroid ABkb, (fig. 9) the cylinder exceeds the
frustum : and the difTerence betwixt them is the same as in the
eoine CT^trd, the plane Drd, or Wib, being supposed parallel tQ
those which terminate the frustum. In different inclinations of
those planes, when tlie altitude of the frustum is given, that
difference is reciprocally as the cube of t\ie diameter B&, which
is the conjugate of CA^ the axis of the frustum. But if the alti*
tude of the frustum be also varied so as to be redprocaUy pror
portional to the diameter 'Rb, then the difference betwixt the
firustum and cylinder shall be always of the same magnitude in
the same spheroid or conoid. When the inclination of the
axis of the solid to the planes thatterminate the frustum is given,,
the difference betwixt the frustum and cylinder, in the same oc
in similar solids, b as the cube of their common altitude. »

The truth of this general theorem will easily tfppear from what
has been demonstrated of these solids, if we prove that it obtains
in the cone. Resuming, therefore, the construction of the sixth
figure, let £B (fig. 1 1), the altitude of the frustum generated by
the trapezium EBCH, be bisected in/; let /a parallel to BC.
meet AC in a, tb parallel to AB passing through a meet BC
and EH in b and e ; and, BR being equal to iZ, or one half
of CZ, complete the parallelogram EBRV. Then, since CZ
is bisected in ^, the rectangle CBZ (or the square of BX)

added

i8 li^Tftddmvtoiri

Added to Ihfe ^iffkife 6f ftZ <lt BR^ In «qufil <8. i. £itcli) fib lltf
square ofTSb; Md ihe eft^tgHSShMi Vjr^imn^Mu^lt Bi
in fh6i«foie equal tb the sum o^ thif eyldidert gAvmied liy thtf
fectengkftBY Artd BY. The frudtuhi c^frt doiie gen^mtcd by thi^
trapezium VSCH id equal to the c^yliftder generateA by the iiso^
tangle BY added to ihe €otie gen^Med by the triangle Ell6, hf
irhat has beeil demoni^ti^ted ; aiid therefofre this firoatuinexceedil
the cyiMer genetated by i!he t-ecifttigle Bn bjr the djteass of th«
coiie getibt^ted by tlie triioigte BBS abdvief the cyttfider gtm^
fated by ihb tectdiigle BY ; that k, (becaifi^ BR is oner half
f)f ]3S> by 6TAt fourth part of liie ddite geherMed by the tiim^
gle EBS. Bill thii excess is dlw&ys of the sdiM ibaghitiide^ hi
th«- saaie or* lA similar coae^^ iVhen EB^the Altitude of tiie frda^
tusakk ii^giv^il : ihetetate, iFa eMeistutby ilueeplaties perpen^^
di<ml)lr to il^ tati^ at equal distanceiiflrOni^ each Other^theftastUfil
comprehended betwixt the first Md th&rd plitnee:s:beeds the cy«^
Muder of &e same height upon tb^ midifie section as its base^ by
k ^milaf tGhe <^ a height e^^ to thi^ ^(Uitade of xke fttn^tdSn;
trhich excfe^ k always of the same Magnttade^ in the siune of
ki Bimikr cbn«i> whe^ tiiat height is giten. In Hke manner^ it
may 1^ demonstfi^ted that if the par^Iel plattes (fig. 12) Cfnd,
Krdt, Huh, atr equal distances from ^ach other^ be obliqile to
fhe axis of the eofte^ but perpendidulal' to die plane ACBc thai
ftasses through the axis ii^ the light lines Ce, Kh, tLki HA ; and
Eddrs be a eone similar 16 ACrnc^ ot Aie same height witii th<l
finistum CfVk : then this frusAMfe shidl exceed the cyUhder ol
li^ eqetBl height on the base JSmk by one fem^ pitft ^ Atf
cone ESks*

^e frustum oftfie^^dg^^tiiiedl^ydiedreaEH tX? CfigM)

h eqnid to the solid generated by the Ir^rpeuum TXZR; 9^4
Ijhfe c5y]ihder generated by fhe rectM^eEMQGiseqtisd to the
sofid gerieratedliy the rectangleTR^^ by whathasbecindemon^
strated above: therefore the differ^ice betwixt thef frustum and
cyUnd^ in the spheroSd is the same as in the con^gt^nerated by
ihe triaAgte C A1>, the altitudes of the f^uslutfis being eqttal.
In the hyperbolic c<moid generated by tke area AIBC (/ig. 10)
l^olving* on. the a^is AC^ the friBtim^ generated by th<(

area

Platell /iz.BliAW./.

^&7
D T V H C

n — >i r jT ^

— /V yr — '
t/ L/ L

^ f G 1

f^.II

:

y

1

/"

*

J

^

r-

n

*• m (if

lllTRO0UCTf«V. 9f

ftfe^ EHL6 U eqqiJ to thfc solid generated by the flapezhut
XXZi*^ aiid tile cylinder geaerated by tbe rectai^ EMQQ
10 equal to tke B^iid geoeiated by the rectangle Ttgm; and
iheiefioore the diffsience betvrbrt the fimitum and cylinder ia thfe
same as m the cone generated by the triangle OCtf revolving cm
the same axiSj the altitudes of the fnistams being eqnal. And
by a similar demonstlraticHi this property isextended toaa^ fm*
turns of these solids temiuiated by partdtel planes obhqiie to
the axis of the generating figure.

BattDietttrnto^rcAimeiJes. HewaBthefintwfadwasabletO
give the exact quadratiire or mensnrafcion of a space bounded
by the arch of a curve and a fight line^ by demoaslratingf that if
ABC(p2.3|/fg. ld> be 1^ segment of apdrabQh^and1>Bpatal«
Id to tite aads of the fignie bisectb^ the base AC in D rneei dsa
cnrve in B^ the segment is to the insoibed triang^ ABC as fims
^ to tbiee^ or is etpHX to the lanangle ACtt^ the j%ht Ihse Cm
parallel to the axis being to BD in that prQt>ortiosi. Hia de^
monstration may be repreaented in the following manner. Let
the base AC by d continaal bi8e<(:U.on be dini^ed iato theeqaal
parts AR, RG, GS, SD, DX, XE, EY, YC, s^i let paraUda
to the axis tlnoikgh AkepOiilts of division meet the curve in P^
H>M^B>N^F^V. Then, if RP pass thl^ough the division of
4ie base that is next to the point A^ and meet Aa in^42» the figmta
APHMBNFVC mscribed in the parabohe segment sh*H be al*
vays ec^oal to the trapeaiam CRda, as we shall dicnr afterwards.
By eobtuming to bisect the pairts of the bose AC^the paintR
thai apfMroaeh to A». and the trapezinm CRd!ix to die triangle
GAa^ so that thor diffeience may become less than my quanti-
ty tintt^anbe assigned. Themscribed polygon APHMBNFVC
^ the same time approaches to the areaof theparaboBe aeg^
mtfat ABC, so that their difoence may also beconte less than
ap^ quantity that is assignable ; fortbe tangent at H being pa-
raUel to AB^ any triangle AHB mni^t be gieatet than half ther
segprnani AHB m iAAch it is inseribed. Therefore^ theparabolic
sflgmenA ABC and thset tiiangkr AaC am eqnal to each other*
F^rif the triangle AoG sfaoold be supposed, to exceed tha para--
hiAcsegiMiit ABC byaoyspace^asO;. then, since by con«

tinning

^ iSTRODVCTIOir.

tinuing the diviaons of the base AC the triangle ARd might
become less than the space O^ the trapezium CRda, by ap»
preaching to the triangle CAa so as to differ from it by a q)acQ
less than O^ might exceed the parabolic segment ABC ; and the
polygon APHMBNFVC, which is always equal to the trape-
zium CRda, might exceed the parabolic segment in which it is
inscribed; which isabsurd. If the parabolic segment should be
supposed to exceed the triangle ACa by any space O ; then,
since by continuing to bisect the base AC the inscribed polygon
might approach to the parabolic segment so as to differ from it
by a space less than O, this polygon might exceed the triangle .
CAa. But the inscribed polygon being always equal to the
trapezium CRda, and the point R being always betwixt A and
C, the polygon must be always less than the triangle CAa ; and
these are contradictory. Therefore the parabolic segment ABC
and the trian^e AaC are precisely equal to each other^ or the
segment ABC is to the inscribed triangle ABC as Ca is to BD,
that is^ as four is to three.

We supposed^ that the inscribed polygon APHMBNFVC ia
alwap equal to the trapezium CRda. To complete the de«
xnonstration^ let DB bisecting AC, GH bisecting AD, RP bi-
secting AG, meet Aa in the points by c, d, respectively ; let GH, >
RP meet AB, AH in L and / ; and let HQ parallel to AC meet
BD in Qy and Tq parallel to AB meet GH in q. The triangle
ABC is equal to the trapezium CDba ; because the triangle ACa
is to the triangle ABC as Ca is to BD, or as four is to three :
and the triangle ACa being quadruple of AS)b, it is to the tra*'
pezium CDba in thesame proportion of four to three. The sum
of the triangles AHB, BFC is equal to the trapezium DOcb ;^
because the sum of these triangles is to the triangle ABC as HL
(or BQ) is to BD, or as the square of HQ (or GD) is to the
square of AD ; that is^ as one is to four : and the trapezium
DGcb is to the trapezium CDba (or the triangle ABC) in the
same proportion. Inlikemanner^thesumof thetriangles APH, •
UMB, BNF, FVC is equal to the trapezium GRrfc ; because
tlie sum of these triangles is to the sum of the triangles AHB,
Bf C as P/ (or H;) is to HL, or as^the square of ¥q is to the-

squax**

. WTKoi^trctiair. f§

«4ii^ie of AL; or as Ike square of R6 is to^the square of AG)
•tiiat is, as one is to four: and the trapezium GR<2c is to the ira-
pesauux DGcb (which is equal to the sum of the triangles AHBj
BPC) in the same proportion. Thus it appears, that the sum
^f the triangka which are subducted from the parabolic seg-
ments, and acUed to the inscribed polygon at every new biseo
jjoa of the parts of the base AC, is always equal to the trape*
xium that is at the same time added to CRiaby iMsecting AB,,
the part of the base neact adjoining to the. point A ; and there^
fore the polygon APHMBNFVC inscribed in the parabolic
aegment islalways eqnil to the trapeaum CRda terminated by
4he gi¥^R r^ht line Ca and a parallel Ri2 drawn through R^
the divi^n of the base next adjoining to the point A. Thia
irapeziam CRda approaiches continually to its limit, the triangle
-CAa ; the inscribed polygon at the same time approaches to the
paraboltc segment, it» limit ; and these are equal to each othefj
as we have demonstrated. After all the methods that have
lieen proposed for danonstrating thequadratureof theparabola.
Ibis, which we bavededcribed from the inyentor, seems to hare
a particular elegance* On this ocgmIou. he shows how to find
the sum of any number of terms that decrease constantly ia
the proportion of four to one ; and, by this example of a geo*
metrical progression (as it is commonly called), opened up a,
subject which has been treated at great length by the modem
Geometriciatts*

Archimeda having demonstrated the quadrature of the pa-
rabola, having also shown how to approximate to the area of
the circle and dlipse, and liow to compare spheres, spheroids,,
and oonoids, or any portions of these solids, with given cylin«^
der» or cones ; there seems to have been nothing neglected by.
him that was necessary to complete the mensuration of all the
%nre8 then received into geometiy, and of the solids generated.
by.tl^ean^ if it was not an^approximatipn to the hyperbolic areas..
But tibis does not appear to have been much considered by Geo*
metricianfl,tiUtheanalogyof theseai^aswith the logarithms wai^
observed ; ^'the solution of the most diiHcult problems of this^
isinA was fouod to depend ou the . measures of arigles or the
» ' areas

36 nvMimrcTmr.

aieas of die fjrofe^ and the measores of nUoi or the a^^
liypesrbola. Beflides what velales to the circle aod. oook we*
tioBB, end Uie Bolidfi generated bf th^>hehas gurenas atiea*
lise coQcerning the qHral line which isdcttcrihed hj a poiiit mov*
iaig with a given vekxnty ^liong & right liae that rerolves at the
aametime with an uni6>i;iB iiagdiar motion about one ctf its ex*
tieieiities. Theproportaofioftheareaof thigear^to the awA
•f the circaBftBcribed ctrde is easily deduced froai the principka
which have been already demcnsl^rated.

m

Let C (fig. 14> be thebeginiiitigof ihtgpiani, CB thentaa^
tionof the revolving ray at theb%iBitti%^f ^ Motiioa; and th€
angles ECA,ACB,BCD,DC£ being aqtud^etCA^CByCD^CE
BieetthespiralinA^B^DaiidE: then, the angular motioa of the
lay^ and the motion of the point that descfibeB the curve along
tfie ray, benigbolh uniibrBi, the right Unes CA^ Cfi, CJ>^ Cfi
must always increase by the same difference cipial to tfie fint
line CA. From the center C describe the tiaiilar arches AQ>
BI,i>L, BN, which meet C£, CA, CB^CDiaO, I, LaivdM
withoat the spiral^ as also the similar arches AH, 3JL, DM that
are aH irithin the spiral, and meet CB, CD, C£ in H^ K ami
M. The spiral area CABDE is alwajs a limit betwixt the sam
of the circQmseribed9ectorsCOA,CIB,€!U),CN£aadtIieaiiaa
oftheitt8cribed8eGtor9CAH,CBK,CDM. The cixcfe describ-
ed with the radius CE is the sum of as many sectass^ equal W
the greatest CNE, as there are sectors ; and therefore (since the
sectors are as the squares of CA, CB, CD, CE) mie thfadpste of
the area of this circle is alsioa limit betwcrt the subbs oCth^i
circumscribed and inseribed sectors. Thsiiill^Brenoeoftiie:
of these sectois is equal to-CNE, which, by {naeeting the angka
at C, may become less than any iqiaoe tfiat can be asBignfit ;
from which it follows, that thespital ai:ea GABD£ is eqpial ta
one third part of the circle of the radius.CE. In like mmmer^
the whole spiraLarea goieraled by th^ragrdniwa fram^thepiMipl
C to the curve while it makes two levohiticatt, is the tfaaidpacC
of a space that is doublec^ the circle descrpfaed with a ladias
equal to 2CE; andthewholeareageaesatedbjrthe raj from tha
beginning of the motion tall after any nnmber of sevobitioas,aa

equal

f^llpl to t)k& t]^s#r4 0^41 fl|i«M;e tbai is the same mvhifle oS
<he Qfiicie ^^poriM^ ^lih. |)^q gne^Ate<t rf y^ 89 themi9^of kh
Kd«lie»yi fe of uoi(. 4py pP^<3kii of t)i^ are» of the spiral tba|

in ihftifUWPWlil^itobdeqnfBl ^.<»^ thi^d part of t)^ aeclQK

«f As WBPkmg x^ when tbe point th^t depcril^es tk^ c|mr#
ifUciaiiiiMi Ca We 9^ ^i^&enm:d0 gi^ «^ Accom^t ^hii
thoorena GOi)fieiiung thf t^ng^ti^ of thifi ci»nre.

* «

. I^;yitt»tek«fc»qoarioPjfioMitheffpjirriqf^r cAi ^
te thaiiriMrh i&deacribed hj fimHftr . w>tiiw& on the sw&ceof •
qaheieyiind fiadaajMtftiQa of tbeauilMie termio^tedby thi^QHrd
to.li^ equfil to Ibe sqiwre of (be ^ajn^etcpr. Let C Qfg. 15) b^
thfi Mister «)f the (fih^i ARBiA a gre^ ciirclej P it« pole;
aii4 «bile Uie qnadrMt PMA je^vQlye^ ^b^t th^ppkP with
9i| wiifoHft m>tiQP» l^t a. p<4»t pY^QQiediQg fiom P move with
%.9M& yeloeity akmgthe quadi:a:»t, and triM)e..i:q[K)i| the sf^^
Oil Mii^Cace ihe ftpkal Wfn. U^i PMA ba ^e situatiw of dm
^fwdm/kt aX tbebi^iiuiing of <hf»iikqi^><Hi» ^mA k^ the point iibak
4B»QOhft».t)ie spiffat oome to a w)ym the qfjudrant comea to 1^
ftoiltioiiPffia. Suppose any tvo qittfluckwt9 PFR, Pfir tomeet
Ihe^ fll^kai ia F,fmd the Qmk AB3 m it aadn; throi^ tf
mdf:6m» the^pole P deapiihe the cirolea MFN9 1/Q» iqm(*
ing^the jyimioirote APB in M>.N, l.^ Q ; let the circle UE^
mmPfria U, amll/Q meet PPR in E;, aj^d fifTdiatt be t«
Jlr aathe <yi94nAt PMA is to tb« t^ah Aa, or aa tb^vdopit;^
of ilia pant, wbidi describe the spiD»l^ alpaig the ygriimg
PFE to the Mriof^ily of Ria tip cirok AR3. Upon CP lei
C^ib0it»kan 40 that the jKii^w^Qf CP m^y he: to tlj^ aipaieqC
C^](i>thMa«epK^rtion. BroinC.«4 center describe the cirabt
pmh in the eawe plane with J^P^; jwdwfi CN and CQ tilt
tbcgr. maet this oirde in n mdq; from the ceQtreij> describo
1too9gb^ 49d 9 the archea nu, qs n^^tin^pq^jm in the pouiti
u and 9. Theo» 9in((^ the are^i P£H is to PHi* aa the sqnare. of
the chord PF is to the square of the chord PR^ and the

r •

^CoIIsct. Matbrak U^. 4. p joy, XU

area

52 riitRODtrcTloir*

atea PRr is to the sector CRr as the square o( Pit is to i!t0
square of CR, the area PFH is to CRr as the square of PF (or
PN) is to the square of CR, or CN. But CRr is to CNQ $0
Rr is to NQ^ and therefore as the square of Cp is to the squaie
ef CP^Joras the square ofpn is to the square of PN ; so that Ibe
Area PFH is to the sector CNQ as the square of |m is to thcf
iftquare of CN. The angle npu being equal to one half of the
angle NCQ, the sector pnu is to one.half of the sector CNQ Iw
the same proportion of the square of pnr to the square of CN s
Therefore the area PFH is double of the sector pnu ; and, in
the same manner, the area PK/*i8 double of the sector pqs. If
^e suppose the quadrants from the pole P to divide the arch A»
into any number of equal parts, and to meet the spiral in any^
fwint^, HsTff, the right lines CN,CQ produced to the circle ptfr
irill' divide the quadrant pnb into the same number of equal
parts: and the spaces PFH inscribed in the area PF/aP being al»
ways double of the sectors pnu inscribed in the segment jpnAp^
the spaces Pl^that are described about the spiral areabei^g-
ttlwaysdouble of the sectors jp^s described about that segmenty
and the difference betwixt the sum of th^ circumscribed and*
inscribed sectors being equal to the greatest sector (which, by
continuing to bisect the pairts of the arch Aa, or quadrant j^ni,
may become less than any assignable quantity) ; therefore the
area PFaP terminated by the spiral PF<z and the quadrant P^na
is double of the segmentpnbp. The arch ARa is to the quadrant
PM A (and the sector C An to the quadrantCPNB) as the squareof
Cp is tothe squareof CP,or as the area Cpnb to theareaCPNB;
and therefore the sector CAa is equal to the quadrant Cpnb^
But it follows^ from what has been demonstrated above, afber
Archimeda, that the surface PMARamP is double of the seo
tor CAa ; and therefore it is also double of the quadrant >Cpn6.
From thissurface takeaway the part PZFii»iiP,andthereremaiii^
the surface PM ARaFZP(bounded by the quadrant PMA, the
arch ARa, and the spiral PZFa) double of the triangle Cp6, and
therefore equal to the square of the right line Cp.

Suppose, for example, the quadrant PMA to make acomplete
Involution in the s6me time th,at the \mxk\ which traces the spi-
ral

I

tKTAOD0CTIOK. $3

tftloA the surface of the gphese describes the quadrant^ which is
the case considered by Pappia. Then^ Rj> being quadruple of
Il^or NQ> Cp must be doable of CP^ and therefore equal to
AB» the diameter of the sphere. The portion of the spherical
siirfeoe terminated by the whde spiral, the drde ARBA and
the quadrant PM A, is equal to the square <^ AB. In any other
case the avea PMAaFzP is to the square of the diameter AB^ la
the same proportion as the arch Aa is to the whole circuui£>
lence ARBA; and this axea is always to the qiherical triangle
PAa^ aa the inscribed square is to the circle.

It is demonstarated in the same manner^ that the area PZFP
IS double of the segment /mti^. But the sector CAR is to the
sector CPN^ aa the arch AR is to the arch PN^ and consequent-
ly^ as the square of Cp is to the square of CP^ or as the sector
qm is to CPN ; so that the sector CAR is equal to cpn : and
the spherical triangle PAR being double of the sector CAR^
it is also double of Cpn. From this ^herical triangle subduct
the ai«a PZFP, and the remainder PM ARFZP must be double
of the triangle Cpn. Therefore the portioa of the sphoricalaur-
lace» terminated by the quadrant PMA^ the arches AR> FR^
and the ^icalPZF^admils of apecfect quadrature^ when the ra-
tio of (^ to CP, or of the arch Aa to the whole circumference,
can be asffgned.

We have now given a summary account of the progress that
was made by the antients in measuring and comparing curvili-
neal figures^ and of the method by which they demonstrated
aU liieir theorems oi this kind. It is often said^ that curve lines
have baenconsideied by them as polygons of an infinite number
of sides. But thb principle no where appears in their writings.
We never find them resolving any figure^ or soUd^ into infinitely
small elements. On the contrary, they seem to avoid such sup-
positions, as if they judged them unfit to be received into geo-
metiy, when it was obvious that their demonstrations might
have been sometimes abridged by admitting them. They con*
sidered curvilineal areas as the liouts of circumscribed or in-
scribed figures of a more simple kind , which approach to these

VOL. L C limiU

54 IKTRODVCtlOir.

limits^ (by a bisection of lines^ or angles^ that is continued at
pleasure)^ so that the difference betwixt them may become less
than any given quantity. The inscribed or circumscribed
figures w^e always conceived to be of a magnitude and nam*
ber that is assignable ; and from what had been shown of these
figures^ they demonstrated the mensuration^ or the propor*
tions^ of the curvilineal limits themselves^ by arguments (d> a&stfr*
do. They had made frequent use of demonstrations of this
kind from the beginning of the Elements ; and these are in a
particular manner adapted for making a transition from right-
lined figures to such as are bounded by cun'^e lines. By ad-
mitting them only^ they established the more difficult and sub-
lime part of their geometry on the same foundation as the first
elements of the science. Nor could they have proposed Ui
themselves a more perfect model.

We have already observed, how solicitous Archimedes ap-
pears to be, that his demonstrations should be found to dq[>end
on those principles only that had been universally received be-
fore his time. In his treatise of the quadrature of the parabola,
he treatsof a progression whose termsdecrease constantlyin the
proportion of four to one, which we expressed by the trapezia
(fig. ld^CDfta,D6c&,GRdc,^c. Buthe does not suppose this
progression to be continued to infinity, or mention the sum of
aninfinitenumberof terms; though it is manifest, that all which
can be understood by those who assign that sum was fully
known to him. He appears to have been more fond of pre-
serving to the science all its accuracy and evidence, than of ad-
vancing paradoxes ; and contents himself with demonstrating
this plain property of such a progression, that the sum of the
terms, continued at pleasure, added to the third part of the last
term, amounts always to four thirds of the first term ; as the
sum of the trapezia CDba, DGch,GRdc, added to one third part
of the last trapezium GRrfc, amounts always to four thirds of
the first trapezium CDba, or to the triangle CAa. Nor does he
suppose the chords of the curve to be bisected to infinity ; so
that after an infinite bisection, the inscribed polygon might be
said to coincide, with the paral^la. These suppositions had been

new

xtew to the Oeooletricians in his time^ and such he appears U>
have carefully avoided.

He has demonstrated many other theorems of thib kind from'
the properties of certain progressions^ the terms of .which cor*
Respond to the circumscribed and inscribed figures : but he
never supposed these terms^ or figures, to increase or decrease
by infinitely small differences, and to become infinite, in num-
ber, that their sum might be supposed equal to the curvili*
neal area, or solid. It was sufficient for his purpose, to assign
a quantity that is always a hmit bet>yixt the sum .of aU the
terms of the progression, and the same sum witlvout one of the
extreme terms ; as the anea, or solid, is always a- limit between
the sum of the circumscribed and the sum of the insqribed
figures, which sums differ from each other in the extreme fi-
gures only. He considered but one decreasing geometrical
}Mrogres8ion, and showed how to find in an arithmetical pro-
fession the sum of the terms, and of their squares only^ Of
late^ other geomeuicol progressions have .been emplpyed suc-
cessfully, for measuring the areas of cwrr^ ; the sums of the
cul^es, and of all the other powers, of the terms in an arithme-
tical progression have served for the same purpose : and each
of his discoveries has produced some extensive theoiy in the
modern geometry.

«

His method has been often represented as very perplexed^
and sometimes as hardly intelligible. But this is not a just cha-
racter of his writings, and the antients had a different opinion
of them.* He find? it necessary indeed to premise several pro^
positions to the demonstration of the principal theorems ; and
on this account his method has been excepted against as tedious*
But the number of steps is not the greatest fault a demonstra-
tion may have; nor is this number to be always computed from
those that may be proposed in it, but from those that are neces-

* Plutarch celebrates the simplicity and plainness with which he treats
the most difficult aadabstrufe questioi^s : 'Ov yof In^ «> ytt^rrua xa^i«

C2 wy

tiiy to malce it fall and conckiaive. Beatcks^ fheUt pt^miiiiBry
{propositions are generally valuable on their own aceonat, ao^
render onr view of the whole subject more clear and compkte.
ill his ti^tase of the sphere and cylinder^ for example^ by his
ckmonstrating so ftdly the mensnratioB of the surfaces and so^
ikts^ generated by the internal and external polygons^ we not
only see bow the surface and solid- content of the q[>here itself is
det^tnined^ but we acquire a more perfect knowledge of thi$
^eoiy^ and of aU that relates to it^ with a satisfaction that we
are s^xsjbk is often wanting in the incomplete demonstrations
of some other methods.

By so many valuable discoveries demon^rated in so accurata
a manner^ and by the admirable use he made of his knowledge
in the celebrated siege of his native city/ and upon other oc«
casions^-l- Archimedes has distinguished himself amongst the
Geometricians^ and has done the greatest honour to this part of
learning. He has not however escaped the cenfiures of some
writers^ who being unskilfol in geometry, and unable to reeonr
cile their own conceits with his demonstrations;, have represent-
ed him as in an error^ and misleading Mathematicians by hia
authority. I But though Mathematicians may be grateful^ au-
thority has not any place in this science ; and no Geometry
cian ever pretended^ from the highest veneration for Archie
medes. Sir Isaac Newton, or others, to rest on their judg»

* How he diiccmcerted all the eflbrtt of two Rooan tnmM» oommiaded b j
tiM Piooonrul Mtreellui and liy Afpiui Chndhu, ia the ■«{• of Synpoie (tiU
the city htiag taken by iuipriie and tieachery, a« end was [put to his Ulc. and
enquiriei at once), it described at len^^th by Polybiufy hiry, Plutargh, t^c. Ho
was called vo7iv/aix«»^ and ixaT^y^^p; and, according^ to Plutarch, acqutr-
od the reputation of more thau human learning-. Meiab of Sjrtaeuie, wili& figumt
that an fojppofed to refer to hit ditooveries, aerre rather to juitify hie oonntiy-
SMD fion the vepioach of iiigratitude which feme have imputed to them, than
to do honour to the inunortal Archimedei. Plantim Sifil, Spanhem^ im orat. I.
Julianim

t Dtodorai Sicului lellt o», (/«^. 6.) that when Archimedet totfelled into £fyptl»
he Invented machinca that were of |^reat uie to that nation^ apd procured him. an
nniverfal reputation.

i Decepit illot auctoritas Archimedit, t^c. ffoBBes Je frrincfftu & rafioefna*
iime Ce^melrarum. The leaned JtMeph Scallgtr tod othen haTt alio wrifteo a^iwt
hia.

ment

inent in a matter of gtometrical demonstration, Tbe pnnnit of
general and easy methods may have induced Bome to make use
of exceptionable pritidples ; and the yast e^teiit which the sci«
ence has of late acquired may have occasioned their proposing
incomplete demonstrations. They may have also sometimes
&Hen into mistakes : bat it wiU be found difficuh to assign oca
false proposition that has been ever generally received by Geo^
metricians ; and it is hardly pos8ible> that aoousations of thk
nature can be more misplaced.

In what Arcfdmedts had demonstrated of the limits of fi*
gures and progressions^ there were valuable hints tofwards a ge^
lieral method ofconsideringcurvilineal figures; so as to subject
them to metumration by an exact quadtatune, an approximation^
or by comparing them with oriiers of a more simple kind. Such
JUethods have been proposed of late in varioas ibrms, and vsgon,
different principles. The first essays were deduced trom a carei»
fid attention to his steps^. But, that his method might ba
more easily extended, its old foundation was abandoned, and
suppoMtions were proposed which be bad avoided. It was
thought unnecessary to conceive the figures Gurcumscrlbed or
inscribed in the curvillneal area, or s<^d> as being abraysassign?
able and finite ; and the precautions of Archimedes came to
be considered as a check upon Geometricians, that served only
to retard their progress. Therefore, instead of his assignable &•
nite figures, indivisible or infinitely small elements were substi^
tuted ; and these being imagined indefinite, or ii^nite, in niUA?
her, their sum was supposed to coincide with the corvilineal
area, or solid.

It was however with caution that these suppositions were at
first employed in geometry by Cavakriut, the ingenious mr
thor of the method of indivisibles, and by others. He disco-
vered a method, which he found to be of a very extensive use,

• C»«toit m ofaTtmnt 4e pi£t la mucfae d'Aichimrfe qu'il [M. ie Roherval] cUit
Wfvi i Mtte lbl»litte it ntrv^iUeufe iciMioe» ^c, Owjrag, ie /V^W. R*^^* 1693.
TIu» M geiMnlly acknowledged by the writew of iliat ti««.

Cs and

38 INTRODUCTIOK.

and of an easy applicationj for measuring or comparing planet
and solids ; and would not deprive the public of so valuable an
invention. In proposing it, he strove to avoid * the supposing
magnitude to consist of indivisible parts^ and to abstract from
the contemplation of infinity ; but he acknowledged, that there
remained some diiSiculties in this matter whidi he was not able
to resolve. Therefore he subjoined more unexceptionable de-
monstrations -to those he had deduced from his own principle ;
and the disputeswbich ensued (the firstof any moment that were
known between Geometricians) Justified his precautions. After-
wards, infinitely small elements were substituted in place of his
indivisibles ; and various improvements were made in this doc-
trine. The method of jirchimcdes, however, was oflen kept
in view, and frequently appealed to as the surest test of eveiy
new invention. The harmony betwixt the conclusions that
arose from the old and new methods contributed not a little to
the credit which the latter at first acquired ; till being more and
moie reUshed, they came at length to be generally admitted oa
iheir own evidence, and seemed to merit so favourable a recep-r
lion, by the great advantages that were derived from them for
resolving' the most difficult problems, and demonstrating the
most general theories, in a brief and easy manna:.

. But when the principles and strict method of the antients,
which had hitherto preserved the evidence of this seienoe entire,
were so far abandoned, it was difficult for the Geometricians to
determine where they should stop. After they Jiad indulged
themselves in admitting quantities, of various kinds, that were
not assignable, in supposing such things to be done as could not
possibly be afiected (against the constant practice of the anti-
ent8)> and had involv^ themselves in the mazes of infinity ; it
was not easy for them tq avoid perplexity^ and 8ometime3 erron

* Qium4 eoQtinui cprnpotitioQeiOy ma&ifeituni est tx praoiteiuU, ad ipium ex indivi-
•ibilitms componendum aoc minimi cogi : aolOm enim omtinui sequi iDdiTisibiUom pro-
|KirtiQoem, ^ ^ oonTcrso, proharv intentum fuit ; qpoA quidem cum atnu|UB poiitione
ftme potest. Tandem Terb dicta^indiTisiUUum afgregata aon ita pcrtractarimw/ ut
infinitatit rationem propter infinitis lixieat wn plm tubtn videiatur, dec* C4Vfikm

or

tKTRODUCTIOir* - 39

or to fix bonxtclfl to* these liberties when they were once intro-
duced. Curves were not only considered as polygons of an in*
finite number of infinitdy little sides^ and their differences de-
duced from the different angles that were supposed to be form-
ed by these sides ; but infinites and infinitesimals were admit-
ted of infinite orders^ every operation in geometry and arith-
metic applied to them with the same freedom as to finite
real quantities^ and suppositions of this nature multipled^ till
the higher parts of geometry (as they were most commonly
described) appeared full of mysteries.

, From geometry the infinitesandinfinitesimals passed intophi-
losophy, carrying with them the obscurity and perplexity that
cannot fail to accompany them. An actual division, as well as
a. divisibility of matter in infinitum, is admitted by some. Fluids
are imagined consisting of infinitely small particles, which are
composed themselves of others infinitely less; and this sub-di-
'vision is supposed to be continued without end. . Vortices are
proposed, for solvitig the phaenomena of nature, of indefinite
or infinite degrees, in imitation of tbtsm&nite^makin geometry;
that, when any higher order is found insufficient for thispurposej
or attended virith an insuperable difficulty, a lower order may
preserve so favourite a scheme. Nature is confined in her ope-
rations to act by infinitely small steps. Bodies of a perfect hard-
ness are rejected, and the old doctrine of atoms treated as ima-
ginary, because in their actions and collisions they might pass at
once from motion to rest, or froin rest to motion, in violation of
this law. Thus the doctrine of infinites is interwoven with our
•speculations in geometry and nature. Suppositions, that were
pr(^K>sed at first diffidently, as of use for discovering new theo-
reins in this science with the greater facility, and were suffered
only on that account, have been indulged, tiH it has become
crowded with objects of an abstruse nature, which tend to per-
plex it and the other sciences that have a dependence upon it.

They who have made use of infinites and infinitesimals with
the greatest liberty, have not agreed as to the truth and reality
they would ascribe to them. The celebrated Mr. Leibniiz

C 4 owna

40 ivTitoDtrcrioK.

owns them to be no more than fictions. OAers place them on
alevel with finite quantities, and endeavour to demonstrate thdt
reality from magnitudes being susceptible of augmentation and
diminution without end, from the properties of die progresuons
of numbers that may be continued at pleasure, and from the
infinity which some Geometricians have ascribed to the hy-
perbolic area. But in these arguments they seem to supppose
the infinity which they would demonstrate.

It was a principle of the antient Geometricians, that any
given line may be produced, and its parts subdivided at plea*
sure : but they never supposed it to be produced, till iti^ould
become infinitely great ; or to be subdivided, tiO its parts should
become infinitely small. It does not necessarily follow, that,
because any given right line maybe continued further^ it can be
produced till it become actually infinite, or that we are able
to conceive such a line to be described, so as to admit it in geo-
metiy. In general, magnitude is capable of being increased
without end; that is, no term or limit can be assigned orsup^
posed beyond which it may not be conceived to be ftirther in-
creased. But from this it cannot be inferred', that we are able to
conceive or suppose magnitude to be really infinite :* or, if we

are

* f tt t late imtin amlbed to teclelmted totlior, Jtistlf tHiMOBi ftr Mt vi.
tfoQi Writiagt, wreial tifiiiiMatf «i» ptopotH; for «dUittin|^ mft^ltudB tetnMjr
iafiaites not that kind irhich hu bp limits, compMkondi sUy «fld €«i xsceive
no ftddition, which he callt metaphysical : hut that which he dafioM to be greater
than any finite magnitadey which he distin|;uiihes from the fonner, and calla
geomttricalm Frntfue U groHdmr tit m§e^Ukie d*sugmenUUim samjim 0m Updmi
^ o m tm k pa tuppoatr migmeniU ime infimU det fritf €*e^-Jk dire qf^Utte term devg^
iMMr infme. £/, «» effete ii est impouikle gue la ^andeur tuuepHble i'tmgmenta*
Hen sans fin soii datu te mime cos fue si elle n* en eioii pas susceptible sans Jin,
Or, si eile ne tetoif pas^ elle demeurermt ioujours fnie ; done etant susceptible
'ii*Mtgtneniaiien tansfis^ elle peui ma demeurerpas toufours fiiie^au^ eefttietfie
000^9 dnem'r t^nie* Elem. 4e la feom. de riafiai, { S9. Beoame ma^tude ie
anteeptiUe of augmentation without end, the author ooncludety that we may fup-
poae it vngmented an infinite numher of timet. But, hf beinf futceptible of aup*
mentation without end« we imdentaad only, that no magnitude can be aaatgned or
ooaofhred lo ipreat but it may be aappoaed to mceifv fitfthef tufmentation, aod
thet a gieater than it may ttill be anigned or conoeiTed. We easily conceive that
a finite magnitede may become gmatet and fieatei without end, or that no ter-

minatioQ

tie nWe <o'joia kifinity to any sopposedkka of « determiinM
tfcmniityy and tovsason oonceraiBg magaitude actualljinfiiiita^
it i$ Mt sardy with that penpicooiy that is reqqmd in gtaam^

try.

tt$aififtiovltimtanbeMBlfiie4«f thelnasMW«4lDkitatty«l^ Wttra d«wi
thndon dearly •oncetTe magnitude inoieued an infinita antlw vi tiaai. Ml.
I«ck adUtoidetffestthit we easily Jimn ail liMcfdMiiaibHy of nttftte
«f wli0w additioa tlien h no appiMok: but lie 4iiltai{;«klM teNriit tkk and tNb
ttei of tn infinite ttnnber; a]id«ibj6ina,t&Bt,liowcliif aaevtf «vr Idanvftheininity
oTniaBibermsylw^lSieiebnofliinc; mow efUent tlun Um atevdity of tke wlualidia
Bk fli HBnttite fluflioer*

The letter pift ^ tlie nrpmeirt tttuMiti to thht *< It if tepMilble tluH iMff.
** nttttde, being;' stiKe|]^IMe of angnieiilatiaii wU ht n t end, ean to in ttoaaaecaib
^ » if H wn not «usce{rtiBie •f augtfi e n t ili oii wMliont end* ant X lit wm adt
** nnceptible of vDgBenhitKMi witheni end* II wnnld Mftialu ilwaytCnite. Ttoni.
*■ Ane, einee it it ftoceptiUe «f M u gi n ert a ttfln iKttovt ^lid, the Mntniy nMt tin
* aHowtd; fhttis, it ney liot atwayv lenmin fiaH«» ot H may toenne infifAe.**
The fcrce nf which mrjument leems to he taken nff, hy eouttderiflg', thM, If nal-
aitode was not atuoeptible of angmentation without end, H Wdsld mi «n!y venate
ilwayi fioihsy hnt there woold neeeetarity hd a t^RBy liinfit, or def^tee nf mannl.
tedn wUch cDHdd nrrer he exceeded, or them tnight ho a pMteal nagnlhidft.
Andy by dlowing that thme it no tneh term ot limit, m^Aitnde Ik Ml atp p q ig d
lo he in the nae caae a* if ft wasnoA Wttac»pinA« of angnentatkA without end, (hongii
We thonld leftneifastt it nay heconie infinite. "Wkki U oppoMto tt» tto auppotii^
aaa|;;nitQdO fuaee]^ble of augtnentatlon wifliOBt end, !• aot die luppottnf H
dwnys finite (ibt finhe'nagnitude is eapahle oTheinir facr aa i e d wittout «nd), huttife
lapposin^ it snieeptlhle of no angnxeBtitiaB at idl, or of an dupaenMioa that haa «
tibit or end.

The series Of inimhen, 1, 2, 3, 4, ^e. in fhelr natufd oider, nay he oes.
tfanied wilhoot end ; and it is said, that ^* we nerer come nearer the end of thn
** plog^resiion how great aoerer the number may he tb which we arriye ; wMdk
*' is a diarac^er that cannot hehnf^ to a series of a finite number of tenns. T3iefe.
** ibre tins natiml teriea has an infinite number of terns." And it is added, ** titt
** though we can go over a finite number of terms only, yet all the terms of tUi
" Infinite progtesiion are equally leaU** But if we canoencefre this series lo hava
wy end, it seems to he evident, thsft we most approach to tliis end as we pta-
oeed Irhu the beginning towatds it; and that, while we adrance, the distandi
tof any tenn from the end must decrease (whether this distance he called finite or
infinite) by the same quantity as the distance from any nibse<ioent term decianei^
or the distance from the beginning of the series increases. If we cannot oonoatVo
^e series to hare an end, then we can hare no ideacf its hut term. If wo sup^
pose this series to be continned to infinity, it irould indeed he absurd, after sucli
a supp e sW on, to say that the number of its tenns is finite. But, in treating thii
science strictly, it may, perhaps, be better to atoidthis supposition* For if it la
only a finite nomber of tenns we can clearly conceive, how shall wo judge of fto
ie«^ of the rest? or wherein shall we place the reality of those which tt isim-
ponible Ibr ns to assign! of whicb two kfaids are said to he in this tano series^

cpch

4S . IKT&ODVCTXOir.

tiy. In the same manner^ no magnitude can be conceii^jo
small^ bat a less than it may be supposed ; but we aie nol thece*
fore ableto conceive a quantity infinitely small. A given mag-

nitttde

•■diiofiiuteiiimioiber; tlie fiftC of whicK are taid to be finite, but iodeianmmUe i
the httefy ictaally iiHuiite.

The ugument fiom the Infiiiity of the- hyperbolic aiee ii much iniisted on.
<< The hyperbolic area (^Elem. de la geom. de Vinfin.pref,) is at really infinite^
<< ai a determiaed parabolic area it two thirds of the circumscribed parallelogiam*
^ It is triiUiii: to «y, that the one can be aetuaUy dcKribed, and the other can-
« not. Geometry is entirely intellectual, and independent of the actual dsKtip-
<< tioB aad eiistence of the figures whose properties it disooren. All that is
^ conoeiTed necessary in it has the reality which it siqiposesin its object. Ther»-
<< fore the infinite which it demonstrates it as real as that which is .finite, (ST^ .*'
And the, learned author, after insistin|r<m this subject, ceocludes, that, <* not to
<* ssoeiTe infinity as it is here repremnted, with all its neoeisary oomequences, vk
^<. to reject a geometrical demonstration ; and that he who rejects one, ought to
<< reject them all." But though the actual deicription of the figures which are
ooosidered in geometry be not necessary, yet it is requisite that we should be able
clearly to ponceiTe that they may exist i and a distinct idea of the maimer how
they mayt be supposed to be described- or generated is necessary, that they voxf
have apdpMe in this science. Principles that are proposed as of the most'cxten-
d?e use, and as the foundation of all ths sublime geometry, ought to be deal
and unexceptionable. If this science is entirely intellectual, or if the reality of its
objects is to.be considered as having a dependence on their being conoeiTcd by
the mind, it would feem, that there must be a difference betwixt the reality of
finite a»igpabU lines or numbers, and the reality we can ascribe to infinite lines or
numbers, which are not assignable, and cannot be supposed to be produced or
genentedbut in a manner that is aHdwed to be ineonoeiTable* - As for what \m
•aid of the prabolic and hyperbolic areas, we can conceire any portum of the
paaboU to be accurately described, and its area to be determined, though no ex-
act figure of this kind should ever exist. YTe can also conceive, that the hyper-
Ua and its .asymptote may be produced to any assignable distance : but we do
not so clearly conceive that they may be produced to a distance greater than wshat
is assignable; and we may well be allpwed-to hesitate at such a supposition to
.strict geometry. Any finite space being proposed, the hyperbolic area (termi-
jMted by the curve, the asymptote,, and a given ordinate) will, exceed it by pro-
ducing the curve and asymptote to an assignable distance $ and there is no assignable
limit in this (as in some other cases) which the area may iu>t furpass in mago
.aitode. Therefore it is said, that this area would be •infinite, if the curve an^
.asymptote could bo infinitely produced. ' But no argument for adoritting magni«
tude actually infinite can he deduced from this, which does not more easily ap-
^pear from hence, that a parallelogram of a given height would be infinite if it
oould have an infinite base; from which it cannot be inferred that such a base or
puallelogram can actually exist. , It is often said, that a rectangle of a given
height on ^n .imaginary base (as the analysts speak) is imaginary : but we can-
not thence infer, that an imaginary line or rectangle can exist. It is not, How

ever.

IVTKODVCTIOV. #3

idtadeuaj be sopposed to b^ divided into any assignable nam^^
ber of parts ; but it cannot therefore be conceived to be divid*
ed into a number of parts ^ppeater than what is assignable. The

parts

tT«r» our intwitioo to miutain tlw impoasibilitjr of infinite iDi|;iiitad0 1 but t0^
ihoir, that luch doctrines are not necenarj ooniequencet of the reoeived principlee
of tUt tcienoe, and not ftij praper to be admitted as the poxaoAwoik of the high

At for the hyperholic areu of a higher kind, which are said to he of i finita
magnitude^ though infinitely produced, the meaning is, that there is a eertain
finite space which such an area never can eqiul, though the cuire sfid its asymp-
toto bo produced never so far; to which however the aiei approeches, to that
the excess of that finite space above it may become less than any tpace that may bo
proposed, by pxoducing the curve and its asymptote to a distance that is assign-
able : as, the sum of the trapezia (^. 13) CViay DGc6, GtiJc, life, that aie do-
termined by bisecting AR continuallj, is always less than the triangle ACtf, bvt
fppioeches to it, so that their di&rence ARd may become less than any given speea
O : or^ as the sum of the right lines CD, DG, 6R, t^c. is always less than CA»
but i^pxoeches to it, so that, b]^ continuing the bisection, the difibienoe AR mejr
become less than any assigned (quantity. But we shaU have occasion to tieat of tbeae
afterwards more fully.

la the same treatise (^ 196.) % proof is offered, to show, that, Intheinfittltt
aeries of numbeia proceeding in their natuni order, there an finite nambeii
whoso squares become infinite, which are called indeterminah&e, and are snp--
posed to occupy the obscure passage from the numbers that are assignable to those
that are infinite. A greatest finite square is supposed in Hum pr o gr es si on, and lo-
prfisrntnil by jm ; all that precede it are finite, and all that follow after it are sop*
poitd infinite. The numben in this progression between n and im^ being lesi
than no, are finite ; but being greater than o, their squares are greater than nm^
and, therefore, by the supposition, are infinite. But how can we admit the sop*
position of a greatest finite square m&iber, such u is here expressed "by tml Tho
sunbei 1S0, being finite, is not the next to it in the progression (which exoeede
it by unit only) also .finite. Should we allow that a finite number becomes in-
finite by adding unit to it, or even by squaring it, how shall we dlstingnish finite
liem infinite } We commonly conceive finite magnitude to be assignable, or to
be limited by such as are assignable, and to be susceptible of further augment*-
tion : and therefore infinite magnitude would seem to imply, either that which
exceeds all assignable magnitude, or that which' cannot admit of any further aug«
mentation; these being direttly opposite to what we most clearly conceive of
^te nugnitnde. But neither of thyie constitute the idea of Infinite maguitude, ae
it must be ondentood in that treatise. The former is applicable to those nnn-
bers which the author calls finite and indeterminable ; which, being su pp o sed to
produce infinite squares, mv>t therefore exceed all assignable numbers wheso
squares are assignable and finite. The latter is ascribed to that infinite only which
he calls aaetaphysical, and excludes from geometry. We are at a loss to form a
distinct jdot pna of finite ittcU ai it it toe imdtistood| and it would seem,

that

parts of a gh«n line may be sopposedtol^coiEitiiiMjfyMMstoi
tin they t>ecome less than any line that isi pfoposed ; mni thtt
is sufficient Ibr completing the dexnonstTidite5<^ the atideiits.

But

tttt Hhit more trt tai fatuity It emplojrel lii paMMtiB^ Inlii U» 4oMiiqt if
fal0iiiiesy It 1)econiei the more abttnise.

A piool b ofibred i {astethn (f 999.^ td iImw, tUt tliHt «n iiite inetloii
ip tlM lerieSf i* 7' 1^ i» !*» ^^« whose iqiuiei become infinitely Uttle in-tt* tiHil
^•^\» ix* "ff* V'« ^he fum of the fint leriei correspondi trtth the uea lueSttd^
ed betwixl t(« canuncn hyperboU and itt uympidts ; and it iM td be (afiaite
irkifi the aeries ii supposed to be continued to infinity. The som of the btter
mtim conaspands with the area of an hyperbola of a higher drder, and Iseaid to te
fiiute, eyen when the serifs is supposed to be continued to ififintty ; betebse then
ii • lianil which this ram can never equal, to which however it centiHililty «p-
jewrhcti as we hare already described. This being allowed, it ii sttppesed £tf«
Her^ thet there is an infinite number of finite terms in the first ptOgtelsiMi ; uM
It is tbeooe demonstrated , that there are finite fracUons in the fint leifet whMt
afMfee beoosne iofinitely Uttle in the second, thus : *< If it shmiM be ptefwoded;
H thet.al) the finite terms in t)M,^st series hare theif squares finite in the seoocul,
** there would be an infinite number of finite ' termr in the leeMid M Hell tt in
^ lbl#ii»i «id the sums pf both would be infiiutes SQ that the eonhety of ea
** wnioqbted tmth, that is universally noeived, Would be dettUUtiited." If we
co«|d «]4ow that them is an infinite number of finite terms in the fint series, thil
Mywfwt «pcht have some weight. But this is a supposition we cannot admit.
Ter the denominator of any fraction in the firrt series is always equal to the
amnbef of teraos from the beginning, and must be supposed infinite when the
iUHBber of terms u supposed infinite { but a fraction that has unit for its ttnme«
liter, asKl i« supppaed to have an infinite number for Its denominator, eamiot be
supposed finite, but infinitely little : so that we cannot Suppose an infinite nombet
^ terns ia the fint series to be finite. It is often said in this treatise, that theie
if e» isfinste number of finite ^rms in the natural series 1, 3, 9, ^ S, t^e. eofW
tiniied i^ sefiiuty. But we are at a loss to conceive how this can be admitted t
maotf in ai^ sueh pregression, the last or greatest term is always e<)ual io the atna*
W el tenM froin the beginning, tod cannot be supposed finite when the ttumbet
of leans is supposed infinite. There is an assignable limit which the sum of the
tpQUB pf the second series never amounts to ; but there is no assignable limit trhioil
the sum ef the first series may not surpass (as we shall show afterwards) ; aAd the mm
€rf the terms of the first is greater than the sum of the corresponding terms of tlM
aeeoad, in a ratio that, by continuing the terms, may exceed any assignable ratio Of
a gsneter magnitude to a lesser : and as this is easily understood and demoostratedf
i^ these is no necessity for haTing recourse to such abstruse principles In eidbt tA
ecceunt for it.

It is of no use to cite authorities on this subject, but as they may Justify «• Hi
^Stabliahiiy so noble a part of geometry for avotditig pfinefplei that are so jsmola
cenlested* What Aristotle taught of infinite magnitude is well known* Mr.
I^eslinitx, who for obvious reasons caimot be auspeeted of any prejudice egiiflii
the dectriae of infinites, expresses himself thus : On t'mianutt dam h9 strUs

Bwii% ii ae lmw i MR t A by tiwfe udM to^t taatte* Ow^ A^rtwne
of oMMb mthe fliHtiil m nnact, thait ^ thflie is mmmAk^ ba^
^ cDttcelmUefai mi f p B w ng — Mrfmitdiy gwat flriafi>i^y (BaaM

'^ pffiwiy from finite to iniiiite isobwure a»d mcMiipielHM»<
^ siUe:'* Mid tihccefiMe it is better Ibriw^iat^
aieieiK?eesgecunetxy,toahetreetfi:Qntfaese«appafl^^ The
ahrtraie cmueqiMiioes, thathsYebfea dbedueed Ifomtfaeniby
ngeasMs «e»^ aaay the^radief eadisQe i» to bewaie q£ admil^
li»giheiiias necenazy primciples ia tiytsscieooe, aDclti> adhere
to its antieiit princ^oks.

Mr. Lo€k, whe wxete his exceHMt eawjr, ^ diart nne jm|^
^' dneover how £8tr ilK pcMivrs of ^ widenrtaodiBig leaek^ to

JEri nomires ^ vonf i rinjinu O^ edi$fmfim detnkr terme^ tmnomhr hrfitd^om

We l»Te. Ka^oinod Hme- remarlu^ at tht desin ef some penoiis<£oi whom«9
have a g'reat irgud, to «how why we liave not followed an author who \u me-
rited $o well of MUlieniatidaiii, wbA wHo on erery oAer ocehrion hn iMee }ullw
If applmaid for Ua dbaa ttd iiMitfct Mgr of e n* rin in g tba 4)ilr«e ceoiaall|k
They wh» iaoated «f isMl«a bef«RB him ptocMM, u he otaoeve^^ wkh a H^
IDOfQimeil which, the cpntemplation of such an object naturally impim : Qtumf
M ^ ettnt arrivi Qajp Ke}» ^m «' arretioii avee um eipece d^rfroy Iff it stdiUe kor^
fewr-^4m rt(^oribit1Hnflmeomm mnmMef€ ^Hi foNbii rtt^tmslS fi^ tfitiok
j^ §mmS^ ^i ^r^fi m Mt^ Th^ ttoiiped when they came «o itfaity wiiiim«i^
•£ hU^ dMp4» and iWQectedi it aa an incompiehemible royitery. He adventarai
fiuther, iD order, to discover the source, and penetrate, into tjie first principles 6t
geometrical truth. Infinity, according to him, it llie |^eat trunk from whidi ite
nriooa hsiaches ate derived, anft to whkh they' all lead* In tJui i^raait fmnOt
hediipiaysa liMte aad finite with a iieedom that puts us hi mind of tHe astiett
Fqet and hi« QndPy whom he r^rctents with the passions of men, and minflei
in their battles.. We doubt not, that, if a full and perfect account of all Hat
is most proAnifid in the 1ii|4i geometry could hare been deduced fiom the doe-
Mte of \aMkHy it miglit him been oq^ected foom this authoK : but our ideu df
inftiiAiQs are too obiotve and unadtt]uate to answer this end ; and there are maagr
things adiwnced by all those who have applied them with great freedom in geo-
metry, that give gound to a remark like to Mr. de St. Evremond's, when ho
(observes, that *' it- it turpriiing to find the antient Poets to tcrupulout to pre.
** terse ptdebility in actipos pumiy huroin, and so ready to violate it in npi^-
*' venting the antient of the Godt." Some have not only admitted in&nitet and
infipitesiinals of infinite orders, but have distinguished even nothings into various
kinds : and if luch liberties continue, it is not easy to foresee what abturditxet maybe
tiMttd(«^^towreri^inwhaiitoaI2edt]lefiibltaiegeometry. >

'^what

48 iVTaoovcTtoirJ

^ what things they are in any degree proportionate, and where
^ they fail us,'' observes, ** that whilst men talk and dispnte of
^ infinite magnitodes, as if they had as compkte and positive
^ ideas of them as they have of the names they use for them,
^' or as they have of a yard, or an hour, or any other determi-
^ natequantity, it is no wonder if the incomprehensible nature
^ of the thing they discourse of, or reason about, leads them
'^ into perplexities.and contradictions ; and theirminds be over-
^' hud by an object too large and mighty to be surveyed and ma-
** nagedby them/' Mathematicians indeed abridge their com-
putationsby the supposition of infinites; but when they pretend
to treat them on a level with finite quantities, they are some-
tiines led into such doctrines as verify th^ observation of thisju-
dicious author. To mention an instance or two : the progres-
sion of the numbers 1, 2, S, 4, 5, &c. in their natural order,
is supposed to be continued to infinity, till by the continual ad-
dition of units an infinite number is produced^ which is conceiv-
ed to be the termination of this series. This infinite number
is supposed to be still capable of augmentation and diminution ;
and yet it is said, '' that it is neither increased nor diminished
'^ by the addition or subtraction of the same units from which
'^ it was supposed to be generated." In a progression of this
kind, the number of terms is always equal to the last or great-
est term, and is finite when the last term is finite. If the num-

ft

her of terms be supposed infinite^ the last term cannot be finite ;
and yet it is said, '' that in such a progression continued to in-
^' finity there is an infinite number of finite terms." It is evi-
dent, that no finite number can become infinite by the addition
of unit, or of any other finite number ; and yet ''a greatest fi-
nite square number is supposed in such a progression, the next
to which (though it exceed that finite number by an unit on-
ly) is supposed infinite." From these suppositions it is info-
red, '^ that in such a progression continued to infinity there are
^^ finite numbers whose squares become infinite ;" though it
seems very evident, that a finite number taken any finite num-
ber of times can never produce more than a finite number. We
may perceive from these instances, that it is not by founding the
higher geometry on the doctrine of infinites we can propose to

avoid

€€
€<
€€

IVTEODVCTIOK. 47

avoid tiie parent inoonsistencies that hovebeen 6l:^eo(edto it;
and «moe aa excellent author, who has always distioguiahed
himself as a clear and acute writers-has had no better mccess in

e8tablishingitonthe8eprincipIessitisbetteFfora8toaY>idlhem«
These soppositions however may be of use, when empbyed with
caution, for abridging computations in the investigatioiof theo*
rems, or even for proving them where a scrupulous exictness ii
notrequired ; and we would not be und^:stood to afirm, that
the naethodsof indivisibles and infinitesimals, by whicl so many
uncontested truths have been discovered, are withouta founda*
tion. We acknowledge further, that there is sometiing mar-
"vellousin thedoctrine of infinites, that is apt to pleasemd trans-
port us ; and that the method of infinitesimals has beei prosecut-
ed of late with an acuteness and subtlety not to beparalleled
in any other science. But geometry is best establiskd on clear
and plain principles ; and these speculations are eer obnoxi-
ous to some difiiqulties. If the greatest accuracy hs been al-
ways required in this science, in reasoning conceuing finite
quantities, weappxehendthatGeometricianscannotie too scru- -
pulous in admitting or treating of iu&iuies, of whia our ideaa
areso imperfect. Philosophy probably will always hve its mys-
teries. But these are to be avoided in geometry : ad we ought
to guard against abating from its strictness and eviddce,the ra-
ther, that an absured philosophy is the natural poduct of a
vitiated'geometry. . .

' It is just at the same time to acknowledge, tha they who
first carried geometry beyond its antients limits, anl they who
have. since enlarged it, have done great service, by lescribing
plainly the methods which they found so advantageois for this
purpose (though they might appear exceptionable ii some re- ,
spects), that others might proceed with the same faiility to im-
prove it. Some of them have been so cautious as to erify their
discoveries by demonstrations in the strictest form ; md others
were able to have done this, had they not chose ratter to em-
ploy their time in extending the science. At first, tie variation
from the antient method was not so considerable, buc that it wis
easy tohave recourse to it, when it should be thought necessay

fortWiatisfainiQAafnidiaiyeqamd a scrapakna exacftiiai»
TheGeooietriGiaiis-iti the aieati time madeggeatia yi t>T en iCPti,
Thef ha( tlie aeeurate method and examples of ApcUmedm
Mbre tfem, by whidi tfaej aughl tiy their discovenei* These
■erved tokecp them Amn enor, aad the nevr methods facilitat-
ed their Mrogxess^ Thu» their vieira enlarged; and proUemi^
'tfiatapptBii^atfiistsightofan insuperable difficulty^wcveafkeD-
wards rea>(ved^ and came at length to be despised as too simple
and easy. The mensuration of parabdasy hyperbolas, spkabof
all the hijher orders^ and of the famous <^cloid, were amongst
the earliet prodoctiona of this period ; some of which seem to
ha^e been discovered by sereral Geometricians ahnoat at the
B«netim< It is not necessary for our poipose to describe moie
particttlaiy what discoveries w»e made by fJMTiisefl^Mess. de
Ftrmmtt act De B^o^Mrved, Gregory k SUk Vincentio^ &c. by
whom thoheoreras of jirchimedes were continwedy and appli^*
ed t^ Ae leosoration of various figures.

The JMkmtHea injinitorwn cf Dr. Waltm was Ae fuHest
taeatiae of his kind tiiat appealed befoie the invenlion of the
meAod of Buxions. Archinudm had eonsideved the auma of
tiieterms itan arithmetical progression^ and of tiieir squares cfo^
¥f (or rathr the lintits of these sums described above)^ these
•eittgsumetnt forthe mensnraiiosiot tne ngures-nelMd examm*
ed. Br. hallis treats this subject in a very general man»
ner^ and asigns like limits for the sums of any powers of tlie
terms^ wheher the exponents be integers or fractions^ positive
er negativst Having discovered one general theorem that in*
chides aH >f this kind^ he then compounded new piogresrions
from varicts aggregates of these tenns^ and enquired into the
sums of tlHpowers of these tenns^ by which he was enabled to
measure acurately, or by approximation^ die areas of figures
widiout nunber. But he composed this treatise (as he telb us)
before he lad examined the writings of Archimedes, and he
pr(^>oses hifdieoremstind demonstrations in a less-accurate form.
He supposesthe progressions tobe continued to infinity^ and in-
vtstigates, fcy a kind of induction^ the proportion of the sum
<f thepowe» ta the product that would arise by taking the

greatest

J

xiTfftOBucTioir. 49

greatest ponver as often as there are tarms. His demonstrations,
and some of his expressions (as when he speaks of quantities
mose than infinite) have been excepted against. But it was not
very difficult to demonstrate the greatest part of hispropositions
in a stricter method ; and this waaeffected afterwards by him-
self and others in various instances. He chose to describe plain-
ly a method which he had found very commodious for discos
vering new theorems ; and it must be owned, that this valuable
treatise contributed to produce the great improvements which
soon followed after. A like apology may be made for otfaera
who have promoted his doctrine since hk time, but have not
given us rigid demonstrations. In general, it must be owned,
diat if the late discoveries wiere deduced at length. In the veiy
same method in which the antients demonstrated their the-
orems, the Ufe of man could hardly be sufficient for consider-
ing them all : so that a general and concise method, equivalent
to their's in accuracy and evidence, that comprehends innume-
rable theorems in a fiew general views, may well be esteemed a
valuable invention.

«

CoMferins was sensible of the difficulties, as well as the ad^
vantages, that attended his method. He q)eaks as if he fore-
saw that it should be afierwBMis delivered in an unexceptionable
form, that might satisfy the mostscrupulousGeometrician ; and
leaves this Goftttofi knot, as he expresses himself, to some^/ex^-
attder. Its imn indeed was soon altered, and many improve-
mentsweremadeby the Madiematidans who prosecuted it since
his time that deserve to be mentioned with esteem. But the
method stillremained liable to someexcepticms, and was thought
to be less perfect than that of the antients on several grounds.

Sir Isaac Newton accomplished what Cavalcrius wished
for, by inventing the method of fluxions, and proposing it
in a way that admits of strict demonstration, which requires the
supposition of no quantities but such as are finite, and easily con-
ceived. The computations in this method are the same as in
(he method of infinitesimals; but it is founded on accurate prin-
ciples, agreeable to the antient geometry. In it; the premisses

VOL. I. D and

10 tirT«ODlTCTto«r.

and condnaioBii aie equally accurate; no qUantitiet ate refeet»
ed as infinitely small^ and no pari of a cnnrek suppoaed to co^
incide ^ih a right line. Hie exceUency of this method Jiaa
not been so fnlly described, or so geneniHy aHetided to, m it
seems to deserve; and it has been sometimes represented aaoiwi
levdinaUtheseie^pectsmtlithemethodofinfinitesimals. The
chief des^ of the follDwing treatise is, to show its advantages
in a clearer and fuller hgbt, lemd to promote the design of the
gteat intentor, hy establishing the higher geometry on plain
principles, perfectly consistent with each other aikd with thoie
of the antient Gedmedricians. ^

The method of deinonstimtion which we make moat xmt of in
this treatise, was first suggested to us from a'partioular attention
to Sir haac Newt<m*s brief reasoning in that place of his prin*
eiples of philosophy where he first published the dements of
this doctrine. After the greatest part of the following traatise
was writteUjWe had the pleasure to obsenFe, thatOeometriciaais
of the first rank had recourse to it long ago onseveral occasions^
as a method of the strictest kind. Mr. de Fermatj in a letter
ibCranendus, and Mr. Hvf/fgesdf k his tforofegftmri . oacij&tf o-
riftm, have employed' it for cximpletiag the demonstrations of
some the(»ems that wteae propoiMd kj GmlUtu^ and prcwpsd
by him in a less aoonmte manner ; and Dr., Abthoie; has 4le*
monstrated by it a theorem concerning the taagenls of earv^
lines. The approbation which it appears to have had irain so
good judges encouraged ns to pabhsh the fbUowing treatise ;
whete it is applied for demonstrating the method of flnxions.
The chief puciuit of Geometricians for some time has been to
improve then* general methods. In proportion as these aie y9l*
luable, it is important that they be established above all excep-
tion : and since they save ns so much time and lafaoiur, we
may allow tliemore for illustrating these methods theraselres.

THE

ELEMENTS

METHOD OF FLUXIONS;

0imoBitnM af t« \lm Munsr of

The Antient Geometricians.

T '

BOOK I.
or T«B rurxiOKS or psomjiTaicai. maovitvsks.

CHAP I.

Of the Grovndt of tlm Method.

^- jL he mathematical sciences tceat of the relations of
quantities to each other^ and of all thetr affections that oafi be
subjected to rule or measure. They treat of the properties of
figures that depend on the position and form of the lines or
planes that bound them, as well as those that depend on their
magnitude ; of the direction of motion, as well as its velocity ;
of the composition and resolution of quantities, and of every
thing of this nature that is susceptible of a regular deti|^i««
nation. We enquire mto the relations of things, rather than

J) S theii

^

52 Tfe Grounds of the Book I.

their inward essences^ in these sciences. Because we may have
a clear conception of that which is the foundation of a relation,
without having a perfect or adequate idea of the thing it is at-
tributed to *, our ideas of relations are often. clearer and more
distinct than of the things to which' they belong ; and to this
we may ascribe, in some measure^ the peculiar evidence of the
mathematics. It is not necessaiy. that the objects of the spe-
culative parts should be actually described, or exist without
the mind ; but it is essential, that their relations should be
clearly conceived, and evidently deduced : and it is useful,
that we should chiefly consider such as correspond with those
of external objects, and may serve to promote our knowledge
of nature.

2. In our pursuits after knowledge, we sometimes consider
things as they appear to be in themselves, sometimes we judg6
of theuLirom their causes, and sometimes by their effects. In
ordinary enquiries, but especially in philosophy, we employ one
or more of these methods, according as we find ground for ap-
plying them. The two last may be no less satisfactory than
the first, when there is a sufficient foundation for them ; and by
carrying our enquiries to the springs and principles of things,
our knowledge of thtJm becomes more perfect, and our views
more extensive. In geometry, there are various ways of dis-
covering the affections and relations of magnitudes that corres-
pond to these general methods oF enquiry. In the common
geometry, we suppose the magnitudes to be already formed,
and compare them or their parts, immediately, or by the in-
tervention of others of the same kind, to which they have a
relation that is already known. In the doctrine whigh we
propose to explain and demonstrate in this treatise, we have re-
course to the genesis of quantities, and either deduce their re-
lations, by comparing the powers which arc conceived to gene-
rate them ; or, by comparing the quantities that are generated,
we discover the relations of these powers, and of any quantities
that are supposed to be represented by them. The power by
which magnitudes are conceived to be generated in geometry,
is motion ; and therefore we must begin with some account of it.

* * Essay coooeroiog tho Haaan IJnAta^xaAwg^ book 3 d^p. 25. § 8«

3- No

£hap. T. Method of Fluxions. 5S

3. No quantities are more clearly conceived by us than the
limited parts of space and time. They consist indeed always of
parts ; bat of such as are perfectly uniform and similar. Thosef
of space exist together; those of, time flow continually : but
by motion they become the measures of each other reciprocally.
The parts of space are permanent; but, being described succes*
•ively by motion^ the space may be conceived to flow as the
time. The time is ever perishing ; but an image or representa-
tion of it is preserved and presented to us at once in the space
described by the motion.

4. Time is conceived to flow always in an uniform coiu'se^
that serves to measure the changes of all things. Wlien the
ipace described by motion flows as the time^ so that equal parts
of space are described in any equal parts of the time, the mo-
tion is uniform ; and the velocity is measured by the space that
is described in any given time. As this space may be conceived
to be greater or less, and to be susceptible of aD degrees of as-
signable magnitude ; so may the velocity of the motion by which
we suppose the space Vo \)e always described in a given time.
The velocity of an uniform motion is the Mme at any term of
the time during which it continues ; but motion is susceptible of
the same variations with other quantities, and the velocity in
other instances may increase or decrease while the time in-
creases. In these cases, however, the velocity at any term ojf
the time is accurately measured by the space that would be de-
scribed in a given time, if the motion was to be continued
uniformly from that term.

5. Any space and time being given, a velocity is determine(i
by which that space may be described in that given time ; and,
conversely, a velocity being given, the space which would be
described by it in any given time is also determined. This being
evident, it does not seem to be necessary, in pure geometry, to
enquire further ^at is the nature of this power, aflfection, ot
mode, which is caHed Velocity, and is commonly ascribed to
the body that is supposed to move. It seems to be sufficient fot
our purpose, that, while a body is supposed in motion, it must
be conceived to have some velocity or other at any term of the
time during which it moves; and that we can demonstrate accu«

P 3 rately

54 The GrounA of the B<yik !<.

rately what are the measures of this velocity bX any term^ in the
enquiries that belong to this doctrine, as will appear in Uie course
of this treatise ; especially suice it is the business of geometry,
as we have observed already^ to enquire into the measures^ ra-
ther than unfold the hidden essences of things.

6. 3ut perhaps this explication will not be thought sufficient^
and it will be required that we should propose a definitipn of
velocity in form. The excellent Dr. Barrow defines it to be
the power by which a certain space may be described in a cer-
tain time. Some perhaps may scruple to ascribe power to a bo-
dy> figure, or point in motion. But it is to be observed, that it
is of no consequence, in pure geometry, to what the power
may be most properly attributed. It is indeed generally allows
ed^ that if a body was to be left to itself from any term of the
time of its motion, and was to be affected by no external in*
fluence after that term> it would proceed for ever with an vni*
form motion, describing always a certain space in a given time :
and this seems to be asufficient foundation for ascribing, in com*
mon language, the velocity to the body that moves, as a power.
It is well known^ that what is an effect in one respect, may
be considered as a power or cause in another ; and we know
no cause in common philosophy but what is itself to be consi-
dered as an effect : but this does not hinder us from judging of
effects from such causes. However, if any dislike this expres-
sion^ they may suppose any mover or cause of the motion they
please, to which they may ascribe the power, considering the
velocity as the action of this power, or as the adequate effect
and measure of its exertion, while it is supposed to produce the
motion at every term of the time. We have observed abeady,
that the principles of this method are analogous to the general
jdoctrine of powers, or may be considered as a particular appli-
cation of it. As a power which acts continually and uniform-
ly is measured by the eiiect that is produced by it in a gives
time, so the velocity of an uniform motion is measured by the
space that is described in a given time. If the action of the
power vary, then its exertion at any term of the time is not
measured by the efS^^t that is actually produced after that term
in a given time^ but by the effect that wouU have been pro*

duced

^l^iiq^KL Method if tlvxiom. us

dueod if iti action had coatinned uniibna from that tena ; and^
in the ssone manner^ the velocity of a variajple motioa at any
givai term of time is not to h^ n^asured by t^i^ gpace that is
actually described after that term in a given time^ but by the
space that would have been described if the motion had con-
tinued uniformly from that term. If the action of a variable
powerj or the velocity of a variable motion^ may not be mea*
<ured in this manner^ they must not be susceptible of any menr.
suration at all. It will appear afterwards^ in the course of thif
treatise, that the other principles of this method correspond with
the plain maxims of the general doctrine of powers that are
em{rfoyed by us on every occasion, and are to be reckoned
amongst the most common and evident notions. There are two
fondamental principles of this method. The first isj that^ when
the quantities which are generated are always equal to each
^Ihffc, the generatipg motions must be always equal* The se-
cond is the converse of the first, that, when^the generating mo-
tions are always equal to each other, the quantities that are ge-
nerated in the same time must be always equal. The first is the
louodation of the direct method of fluxions ; the second, of
tbe inverse method. But it is obvious, that they may be con*
9idered as cases of these two general principles : when the ef-
fects produced by two powers are always equal to each other,
then (supposing that no other power of any kind afl^ts their
operations) these powers must be supposed to act equally at
any term of the tiine ; and^ conversely, when the actions of
two powers are always equal tP each other at any term of the
time, then the effects produced by them in the same time must
lie always equal.

7- This method is so well founded, that its rules ai^ opera«
tions may be delivered in away consistent with any general priur
^ples that are not repugnant to the most evident notions ; though
it is impossible for us, in treating of it, to keep to expressions
tiuitxftay q>pear equally consjsl^t with every schemeof meta«
physiic^t Ithas been frequently considered in a manner a^eea*
ble to the principles of those who suppose quantities to consist of
indivisible or infinitely small elements. We are to proceed up^
on more strict and rigid principles : but it will be har.dly pqssiljle

P4 for

56 The Grounds of the Book t .

for us to avoid always such expressions as may hatt been noaie
time or other matter of dispute amongst philosophers. Tbeir
controversies concerning continued and discrete quantity have
not been thought to weaken the evidence of the common geo-
metiy. Nor can their disputes* concerning motion afiect the
certainty of this method ; since we have occasion in it for no
more than the most obvious notions of space^time^ motion, and
Telocity, that cannot be said to yield in deamess and evidence
to the principles of the common geometry.

8. When we suppose that a body has some velocity or other
at any tenn of the time during which it moves, we do not there-
fore suppose that there can be any motion in a term, limit, or
moment of time, or in an indivisible point of space : and as WQ
shall always measure this velocity by the space that would be
described by it continued uniformly for some given finite time^
it surely will not be said that we pretend to conoeive motion or
velocity without regard to space and time.

9* But to proceed : when any quantity b proposed, all
others of the same kind may be conceived to be generated hem
it ; such as are greater than it, by supposing it to be increased ;
such as are less, by supposing it to be diminished. In the com-
mon arithmetic, integer numbers are conceived to beproduced
by adding agiven quantity or unit to itself continually; andfiac*
tions are produced by supposing it to be divided into such parts
as by a like addition would generate the given quantity itself.
But in geometry, that all degrees of magnitude may be produc-
ed, and in such a way as may found a general method of de-
riving their afiections from their genesis, we conceive the quan-
tities to be increased and diminished, or to be wholly generated
by motion, or by a continual flux analogous to it. The quan-
tity that is thus generated is said to flow, and called a Fluent.

10. Lines are generated by the motion of points ; mxf^ces,
by the motion of lines ; solids, by the motion of surfaces; angles,
by the rotation of their sides : the flux of tim6 being supposed
to be always uniform. The velocity with which a Une flowBj

* Oe oat\m moUU k ipcU definitioofl, de csotu ae diilerantiu, eonplun cubtilitet
vsptantur Phytic! ; quinmi fert Mathematicii nihil ooidi vel cum ; fUftOAiiB potest
^^Dcccammniiiewmapioicit, fiarnw, Itet. fgnh l^

10

Chap. I. Method oflhmam. 57-

is the 9tme as that of the point which is supposed todeseribe or
generate it. Ihe velocity with which a sbifaoe flows^ is the
same as the velocity of a given right line> that^ by moving paral-
lel to itself, is supposed to generate a rectangle which is abraya
eqaal to the surface. The velocity with which a solid flows^
is the same as the velocity of a given plain surface, that, by mov-
ing parallel to itself, is supposed to generate an erect prism or
cylinder that is always equal to the solid. The velocity with
which an angle flows, is measured by the velocity of a point,
that is supposed to describe the arch of a given circle, which
always subtends the angle, and measures it. In general^ all
qnantities of the same kind (when we consider their magnitude
only, and abstract from their position^ figure, and other aflbe*
tions) may be represented by right lines, that are supposed to be
always in the same proportion to each other as these quantities.
They are represented by ri^t lines in this manner i^ the £le«
ments, in the general doctrine of proportion, and by right lines
and figures in the Data of Euclid.^ In this method, likewise,
quantities of the same kind may be represented by rig^t line%
and the velocities of the motions by which they are supposed t0
be generated, by the velocities of points moving in right lines.
All the velocities we have mentioned are measured, at any term
of the timeof the motion^ by the spaces which would be describ-
ed in a given time, by these points, lines or surfaces, with their
motions continued uniformly from that term.

1 1 . The velocity with which a quantity flows, at any term of
the time while it is supposedto be generated, is called its Fha-'
ion, which is therefore 'always measured by theincrement or de«
crement that would be generated in a given time by this mo-
tion, if it was continued uniformly from that term without any
acceleration or retardation : or itmay be measured by the quan-
tity that is generated inagiven time by an uniform motion which
is equal to the generating motion at that term.

12. Time is represented by a right Une that flows uniform-,
ly, or is described by an uniform motion ; and a moment oc
tennination of time is represented by a point or termination of

* ScotjMpuQ&c^toUieiWtfbyMariniu^Deaitheend.

that

M Jif^ GfMnA tf tie .Book!*

tkat liiie* A given velocity ia represented by a given line^ the
aeme trhicb wottU be described by it in a g^ven time. A velo^
city that ia accelerated (»r retarded^ is represented by a line that
iiicreaaes or decreases ip the same propor ti<Mi. The time of any
]|iotion being represented by the base of a figurcj and any part
of the time by the corresponding part of the base ; iftbeordi**
Bate at any point of the base be equal to the space that would
be described, in a given time, by the velocity at the correspond-*
ing term of the time continued uniformly, then any velocity
will be represented by the corresponding ordinate. The flux*
ions of quantities are represented by the increments or decre*
ments described. in the last article which measure them ; and^
instead of the proportion of the fluxions themselvesj we may
always substitute the proportion of their measures.
. 13. When a motion is uniform, the spaces that are described
by it in any equal times are always equ^. When a motion is
perpetually accelerated^ the spaces described by it in any equal
times that succeed after one another, perpetually increase*
When a motion is perpetually retarded, the spaces that are
described by it in any equal times that succeed after oi^e ano*
ther, perpetually decrease.

14. It is manifest, conversely, thatt if the spaces described in
any equal times are always equal, then the motion is uniform.
If the spaces described in any equal times that succeed after one
another perpetually increase, the motbn is perpetually accele-
mted : for it is plain, that, if the motion was uniform for any
time^ the spaces described in any equal parts of this time would
be equal ; and if it was retarded for any time, the spaces de«
scribed in equal parts of this time that succeed after one ano-
ther would decrease : both of which are against the supposition.
In like manner it is evident, that a motion k perpetually re^
larded, when the spaces that are described in any equal time»
that succeed aftar one another perpetually decrease. The fol-
lowing Axioms are as evident as that a greater or less space is
described in a given -time, according as the velocity of the mo*
tion is greater or less.

AXIOM

Chap.*!. Mitioi tf Fhxiam, M

AXIOM I.

15. The ipvu de^cfU^Ml Ihf an Atcdertaei ntdtion i$ greater
than ih$ ipau wkM wouid kOve htun descfihei in the same
time, if the motion had not been accelerated, but had conti'
nued uniform from the beginning of the time.

AXIOM IL

The epace described by a motion whiU it ii accelerated, U
less than the tpace which is described in an equal time by the
motion that i$ acquired by that acceleration contimted uni*
fomdy.

AXIOM III.

!I%e 4pac« described by a retarded motion u hm than
the sfoee which would have bem described in the same time, tf
the metien had not been retarded, hut had emUinaed un^orm
from the begimuf^ <lfthe time*

AXIOM IV.

Ths spatt described by a motion^ while it i$ retarded, is
gnstter than the space which is described in an equal time by
the motion that remdins after that retardation, continued
uniformly.

*

16. Before we proceed to etxqtiife into the fluxions of quan-
titiesj it IB necessary to premise the following general theo*
rems^ which contain the grounds of this method. The two
first are from the treatise of Archimedes con<

THEO-

60 The Grounds of the Book L

THEOREM I.

The spaces described by an uniform motion are in the same
proportion to each other as the times in which they are
described.

Suppose a point to describe the right line AB with aniiniform
motion. Let it describe the space CD in the time FG, and the

^pace DE in the time GH ; then shall CD be to DE as FG is to
GH.

Let KD and MG be any equimultiples of the lines CD and
FG ; and let DL and GN be any equunultiples of DE and

GH. Becausef

^ ^ C D. E LB the motion of

the point in
M F G H N the line AB is

supposed to be
uniform, and
it desoribes CD in the time FG, it will describe any line
equal to CD in a time equal to FG, and it will describe KD,
tbatisamultipleof CD,in the time MG, that is an equimultiple
of FG. For the same reason, it will describe DL in the time
GN. But, when a motion is uniform, a greater q>aceis alwajs
described in a greater time : so that, if the space KD exceed
DL,the time MG will exceed GN; if KD be equal to DL,
the time MG wiU be equal to GN; and if KD be less than
DL, the time MG must be less GN. Therefore (by def. 5.
lib. 5. Elem.) CD is to DE as FG is to GH ; that is, the
spaces described by an uniform motion are in the same pio-
porUon to each other as the times in which they are de-
scribed.

THEOREM H.

17. The spaces described by an uniform motion are to each
other in the same proportion as the spaces described in the
tame timet by any other uniform motion,

Suppose

Chap. I. Method ofThacum. 6l

Suppose two points

to describe the right A C . D E B

lines AB, KL with "~ -

any imiform mo- K F G H t

tioDs. . Let CD and — ^— — —

FG be described hj

them in the time M N R

MN ; and let D£

and GH be described by them in the time NR. Then^ by the
first theorem^ the space CD is to the space DE as the time MNT
is to the time NR ; and FG is to 6H m the same ratio of MN
toNRV Therefore CD is to DE as FG is to GH ; thatis^the
spaces described by the first niotion are in the same proportion
to each other as the spaces described in the same^ or in equal
times^ by the second motion* The spiral of ArcMmeda be-
ing described by the composition of two uniform motions^ one
of which is rectilinea)^ the other circular, he had occasion^ in
demonstrating its properties^ to make use of no more of the doc-
trine of motion than these two theorems. But^ in establishing
a general method for discovering the properties of curvilineal
figures^ we haye occasion also for these that follow.^

THEOREM m.

IS. If the spaces that are described m the same time by two
fnotions, uniform or variable, be almays equal to each other,
the velocities of these motions must be equal at any term of
the time.

Suppose the points P and p to describe the lines AK^ ak in
the same time^ with motions uniform or varied at pleasure^ but
so that the space described by P be always equal to the space
described by jp intiie same time. Then shall the velocity of /»
at any term or moment of time be equal to the velocity of P
at the same term or moment. This theorem is so evident, that
it may seem to need no proof. If AK and ak be right lines,
and we suppose aft to be placed upon AK, the point p will be

alwavs

62 The Grounis of the BookiL

^ways oyer the pomt P> and their vdocitieB at the same tenn
df the tijM <smnof hnt he fiuppoaeiLeqM^ whether the motions
hie uniforoi or yariable. But as this is a ftmdamental theorem
in tiik dpctrinej and holds whether Ihe points P and p move
in right lines or in carves, we shall demonstrate its various ctaes
from thepreceding axioms.

19. Cosel. Supposethe motion pfPtpl>e uniform. Then,
Wice equal spaces are describe by P iji j^ny equal times, it

foQows, from the
AI^BMD Jj G K supposition; that e-

qual spaces are al-
m p b m d / g k ^ described by p

in any equal times,
n g H Q N and that its mo-

tion isalso uniform.
But the velocities
of imifarm fnotions ere ^qual when eqiial spaces are described
hy them in the some tiioe ; 9Ji^ theref$>re di/& velocities of P
Md j^ aie in this cam ahrajTi eqi^ to each other.

20. Case £• Suppose that the motion of P is perpetually ac-
celerated, or that the spaces desaibed by it in any equal parts
of time that succeed after one another, perpetually increase.
Then the spaces described by jp being fh^&ys equal (by the sup-
position) to the spaces described in the same times by P, the
qpaces diMcribed by |» jugi any equal parts of time that succeed
ckch other muat also perpetually increase, and the motion ofp
must be peipetuidly aopelerated'(by art. 14). Let P come to
D^ and p to d, at the same term or moment of time ; and their
velocities at that term shall be equal. For, if tbey are not
equal, suppose first the velocity of p to exoeed the ^elocily of P;
let DG and dg be any equal spaces described after that term by
the points P and p in the time lepnsented by HN. Then^
because the motions of these points are^perpetuaily accelerated
during the time HN3 it follows, from the &st axiom, that the
space which would be described in that time hyp with its mo»
tion at d continued uniformly, is less than dg, which is describ-
ed by it with an accelerated motion in the same time ; and it
follows from the second axiom, tliat the space which would be

describ-

Qicp^i. Method of Fkudam; §B

4icaCf^M by the poinft P widi the oioticw it has aeqoiied «t 6
contiiHwd ttflif^rnly far the tkone HN> ii gieaAertfian BO^ wiilch
isderaribed t^ Uio th^ faaa^ time before that moiioB it aoqkw
ecL Thet^ate, mxoe dg k ^qual to I>6 (by the snppouttoti)^
the veh>city ofp Btdklm than the Ydk)city of Pat 6. B«t
the velocity of p at ^ is fiu{)|>06ed to be greater than the Telodr
^ofPatD; and therefore it may be inppoied eopial to the ve^
locd^of Patsome miennediate tarn of the time HN> as when
it oomes to a point h betwixt J} and G, .I^et dl be equal (iq
DL^ ati4 let HQhe the tkae ki which P aiidp deacribe ti|t
equal spaces I>L and dl wUh 4heir acoderatad motiona. Thesi
by the. Sxei axieon^. the apace which would be described in the
time HQ by the miction o£p at d contiiiued uniformlyi is lev
than dl, which is desoribed in the aame time by its accelerated
moUoA ; .and^ by the second axiom^ the spaoe which would ba
described in the same timeUQby the paint Pwith themotiooit
has acquired atLcontitzuedunifonidy^isgreaterthanDL, whiok
was described by it in the same time before that motion waaao*
quired. But by the suppooiion, d/ is ecpial U> DL ; and tlMsie^
fore a leas qpace would be deseribed m tbe same time by tha
moiicm of p at d continued mifyoBiy, than by the motion of
P at L continued uniformly : so that the velocity of p atd
must be leas than <iie velocity of P at L. Suf these velocities
wtre supposed equal ; aiul these being contradictoiy, it appeails
that the vdocity 0(p %i d is not greater than the velocity of
P at D« In the sau^ imanner it is diown^ that the velocity of
P at D is not greater than the velofrity ofp at d. Therefore
the veloci^s of P and j? are equal at this or any other term ef
the time <^ their motion.

21. The velocities with which the points P and/i come to
I) and d may be shown to be equals from the same princifdes,
without supposing their motions to be continued after the term
H^ by considering the spaces described by them before that
term. Let BD and bd be any equal spaces described by the
points P and p with their accelerated motions in the timenH^
before they come to B and d ; and if their velocities be not
then equals let the velocity ofp at d exceed the velocity o^*
P at D. By the second axiom^ the 8j>ace.tliat would be describ-
ed

64 The Grmnds of the Bookl*

td by the motion of P at D continued nniformly in th^ time
nH is greater than BD. By the first axiom, the space, that
would be described in the same lime nH by the motion ofp at
b continued uniformly, is less than bd or BD. Therefore the
Telocity of P at D is greater than the velocity ofp at b : but
it is supposed to be less than the velocity ofp at d ; and conse-
quently, it may be supposed equal to the velocity ofp at some
intermediate term q of the time nH, when p comes to some
point m betwixt b and J. Let MD be equal to tnd, and the
Bpaces-MD, md will be described in the same time^H by the
points P and'p. By the second axiom, the space which would
be described in the time qithy the motion of P at D continued
uniformly, is greater than MD. By the first axiom, the space
wfaidb would be described in the same time ^H by the motion
clip at m continued uniformly, is less than md or MD. There-
fore the velocity of P at D is greater than the velocity ofp at
m: but they were supposed equal, and these are contradictory.
It iqppears therefore, that the velocity ofp at (2 is not greater
than the velocity of P at D ; and in the same manner it is shown,
that the velocity of P at D is not greater thanjJie velocity of
p Bl d; and therefore these velocities ibust be equal to each
other.

• 22. Case 3. Suppose that the ipotion of P is perpetually re-
tarded, or that the spaces perpetually decrease which are de-
scribed by it in any equal times that succeed each other. Then,
by the supposition, the spaces described by p in any equal suc*-
ceeding times also decrease, and its motion is also perpetually
retarded. If the velocity of P at D be not equal to the velo-
rfty ofp at d, let it first be greater. Suppose D6 and dfg to be
equal ^aces described, in any time HN, by the points P and
p with their motions continued after the term H, at which they
are supposed to come to D and d. Because their motions are
perpetually retarded, it follows from the third axiom, that the
space which would be described, in the time HN, hyp with its
motion at d continued uniformly, is greater than <^, which is
described in the same time byp with a retarded motion ; and
it foDows irom the fourth axiom, that the space which would
be described by P with the motion that remains at G continued

uni-

r

Chap. L ^ Method of Utt^tiom. 63

miifonnij for a time equal to HN, is less than DG (or dg\
which is described in the time HN before the motbn of P is
reduced to the velocity which it retains at the term G. There-
fore the velocity of jn at
i^is greater than the ve- A P B D L

lodtvofPatG: and "^ — "" — '

since it is supposed less a p ft d I

than the velocity of Pat

G

g

D, it may therefore be H Q N

suppoaed equal to some """

intennediate velocity of

P, a« to that< with which it comes to some point L betwixt D
and G. Let <tf be equal to DL ; and let these spaces be describ-
ed by the points P andp with their retarded moUons in the
time HQ. Then, by the third axiom, the space that would be
described in the time HQ by the motion of/, at d continued
nmformlyis greater than dl; and, by the fourth axiom, the
space Aat would be described in the same time HQ by the mo-
tion ot P at L continued uniformly is less than DL or dl.
Therefore the velocity of p at rf is greater than the velocity of
P at L : but they were supposed equal ; and these are contra-
dictory. It appears, therefore, that the velocity of P at D is
not greater than the velocity ofp at rf ; and since it is shown
m the same manner, that the velocity ofp at d is not greater
than the velocity of P at D, these velocities must therefore be
equal. It is easy to show tiiat Uiese velocities are equal by
considering the spaces BD and bd that are described by P and
p beforethey come to D and d, without supposing tiieir motions
to be continued after that term.

23. If the motions of the points P andp are sometimes acce-
lerated, and sometimes retarded, then DGand dg, brBDand bd
must be supposed to be spaces described by them, eitiier while
Uiey are accelerated only, or while they axe retarded only We
bave chiefly in view, in these tiieorems, such motions as are uni-

tonn,andsuchasareincreasedordimini8hedbyaconti«uedacce-

teration orretardation. Butthedemonstration maybe extended

to th^casesalso, where tiie motionsare supposed to be increa.s.

VOL I ^ a* certain terms of the time by assignable aug-

Od The CrowA nfth^ Book I.

]](ienU or deoremeiiU at once, by jBvppoung BP aad D6 to be
tjbe f paces described iQ the intervid of tipe betwixt two such
QzcGc^ing leimsj or in a part of that intenraL Qr^ in the flOth
article^ instead of supposing^ in the latter psgrt of the d^nonstra*
tion^ that the velocity ei p dXd\% equal to the velocity of P
at It, some intermediate place betwixt D and O^ we niay sup-
pose thatit iseither equal or greater than the velocity of P at L.
In the last article^ we may suppose the motion of p at 1/ to be
mthei* equal to the motion of P at L, or less thanit; andthusthe
demonstration may be accommodated to those cases^ whan tha
motions of P and|? are perpetually increasing^or decreasing, but
not in a continued manner* See below, art. 44» 45, and 46«

THEOREM IV.

^4. If the velocities of two motions are always equal to each
other, the spaces described by them in the same time are always
equal.

Suppose the points P and jp to describe the right lines AG and •
ag with any motions, uniform or variable, but so as that the ve«
locity o( p at any term or moment of time be always equal to
the velocity of P at the same term. Let X)G and dg be any
spaces described by them in the same time HV \ and DG shall
be always equal to %.

CW 1. If the motions of the points P andp are uniform, it
is evident, that, since these motions are equal (by the suj^iod-
tion), the spaces described by them in the same time must be
equal ; and therefore in this case PG is equal \Jodg*

25. Ctise £. Suppose the motions of the points P and/' to be
perpetually accelerated while they describe the spaces DG and
<4r ; s^d if £1^ be not equal to DG, let it first be equal to any
line DK less than DG« Let the right Une HV, which repre-
sents the time, be divided by a continual bisection into equal
parts, HR, RS, SQ, QV, till the time represented by one of
these parts, as QV, be less than the time in which the point P
describes KG. Let DL, LM ,MN^ NG be the spaces, descrih*
ed by P; and dl, Im, mn, ng the spaces desaribed hj p^ in the

equal

r

Chap. I. Mciiod of Fbixhns. OT

tequid parte of the time repi%aented by HR, RS, SQ, QV ; and^
by the suppoftition^ the vdocitiefl of P at thcf potats L» M^ and N
will be respectively e*

qualtotheveJocitiesof A PPL M K N G
p at the points I, m and

n. The ^eeftg^ which a, p d I m n g

is described byp in the

time QV with an acce- H R S ^ Q V

leraied i^otion^is great*
or than the space which

wnould be described in the same time by the motion of p atn
contiaued uniformly^ by the first axiom. The velocity oip at
II is equal to die velocity of P at N ; and^ by the second axi*
em, the space which would be described in the time QV by
the point P with its motion at N continued uniformly^is greater
than MNy which was described by it in an equal time before
its velocity at N was acquired. Therefore ng is greater than
MN. In the same nmnner it appears^ that mn is greater than
LM, and Im greater than t)L ; so that Ig is greater than' DN,
and ^ IB sorely greater than DN. But BN is greater than DK ;
for the time QV is supposed to be less than the time in which
KG is described by the point P ; so that NG being' described in
the time QV^ it must be less than KQ, and BN must be great-
er than BK. Therefore dg being greater than BN^ it must'be
greater than BK : but it was supposed equal td BK i and these
are contradictory. It appears^ therefore^ that the space ^ is
not less than BG. In the same manner it is shown^ that BG is
not less than dg ; and therefore these spaces BG and dg, whidh
are described by the points P andp in the same time^ are equal
to each other.

26. Cases, Suppose the motions ofthe points P and/7 to be per-
petually retarded ; and if d^ be not equal to DG, let it first be
equal to a line KG less than BG. Let the time H V be divided by
a continual bisection into the equal parts HR^ RS, SQ^ QV^ till
the time HR be less than that in which Pdescribes-BK. Let BL^
LM^ MN^ NG be the spaces described by P^ and dl, Im, mn, ng
bethespaces described by p, in the equal times HR,RSj SQ,QV.
Then BL shall be less than BK, because it is described by the

£ 2 point

eg The Gromids of the Book L

poiiit f In a lew time ; and die velocities of P at the points L,
M, N, and G will be r€3l)ectively equal to the velocities ofp at

the points I, m,
A P ^ D L K M N G 7 1, and g, by the

" "^ ' \ supposition. The

a p d I ' fti H g motions of the

" points P and p

H R S Q V being perpetu-

ally retarded, it

follows, from
tlie fourth axiom, that nm is greater than the space which would
be described in the time SQ (or QV) by the motion ofp at n
continued uniformly. But the velocity ofp at n is equal to the
velocity of P at N ; and, by the third axiom, the space which
would be described in the time QV, by the motion of P at N
continued uniformly, is greater than NG, which is described
by P with a retarded motion in the same time. Therefore mn
is greater than NG. In the same manner it appears, that Im is
greater than MN, dl greater than LM ; and, consequently, that
dn is greater than LG : so that dg is surely greater than LG,
and therefore greater than KG, which is less than LG. But
rftr was supposed equal to KG ; and these being contradictory,
it follows, that dg is not less than DG. In the same manner it
appears, that DG is not less than dg ; and therefore the spaces
DG and dg are equal. By joining these cases together, the the-
orem is demonstrated, when the motions of the points P and p
are sometimes accelerated, and sometimes retard^.

THEOREM V.

d7- fVhen the spaces described by two motions in the same time
are always in an invariable ratio to each other, the velocities
of these vi/otions are alvdays in the same invariable ratio.

Let DG and dg be any two spaces described by the points
P and^ in the same time ; and let DG be always to <% as £ is

to

Chap. r. Method of Fluxions. ' 6^

to F. Then the \elocity of P at any term of the time shall bt
to the velocity of p at die same term in the same invariable ra-
tio of £ to F. In the first place, if the motion of P be uniform,
so that equal spaces be described by it in any equal times,'
eqaal spaces will be also described by p in any equal times, and
its motion will be ako uniform. But the velocities of uniform
motions are in the same proportion as the spaces described by
them in the same time ; and therefore, in this case, the veloci*
tyof P is to the velocity ofpas D6 is to dg, or as j^isto F.

£8. Case £• If the motion of the point P be perpetually ac-
celerated, then (because the spaces described by P andjp in the
same time are always to each other in a given invariable ratio)
the 0paoes described by p in equal times perpetually increase,
and its motion is also perpetually accelerated. In the same time
that the points P and p describe the spaces DG and dg with
their accelerated tnotions, the poiiit P with its motion at 6 con-
tinued uniformly would describe a greater space than DG (by
the second axiom) ;,and the point p with its motion at d con-
tinued uniformly ^tov^A describe a less space than dg,by the fiist
axiom. Therefore the velocity of the point P at G is to the
velocity of p at d in a greater ratio than DG is to dg^ or £ is
to F. From which it follows, that if the velocity of P at D
was to the velocity ofp old in a Jess ratio than that of £ to
F, then the velocity of P at some intermediate term betwixt D
and G might be to the velocity of /> at d in the very same ratio
as £ is to F. But this is impossible; for supposing L to be such
a term, let the space DL be to ctf as £ is to F, and these spaces
will be described by P and j> in the same time, by the supposi-
tion. It foUowB, from the second axiom, that the point P with
its motion at L continued uniformly would describe a grea^ter
space than DL, in the same time that it describes the space DL
with its accelerated motion ; and, by the first axiom, the point
p with its motion at d continued uniformly would describe a
less space than dl in the same time. Therefore the velocity of
P at L is to the velocity of jp at J in a greater ratio than DL
is to <{/, or £to F; so tiiat the velocity of P at Lmight be to
the velocity of jp at d in the same ratio as £ is to F, and !n a
greater ratio at the same time ; which is absurd. It appears,

E 3 there*

70 The Gr^wOs of the FookL

tlicrefim^ that die ratio of the velocity of P at I) to the velo*
city of |i at d cannot be less than that of £ to F. Suppose now

that ratio to he greats
A P D L G er than the ratio of

£ to F. In the same
a p d i g time that P andp de*

scribe D6 and Jg

£ p with their accelerate

' ' ' ed motioiiSy the point

P with its motion at
D continued uniformly would describe a less space than DO
(by the first axiom)^ and the point p with its motion at g con-
tinued uniformly would describe a greater space than dg^ by
the second axiom : so that the velocity of P at D is to the ve-
locity ofp at g* in a less ratio than D6 is to ^, or B is to F.
Therefore, if the velocity of P at D was to the velocity of p
Bid in 9k greater ratio than that of E to F, th^ velocily of P
at D might be to the velocity of p at some intermediate term
betwixt d and g in the same ratio as E is to F. But this is im^
possible. For supposing / to bo such a term, azkd DL to be la
dl in the same ratio as E is to F, these spaces woaM be describ-
ed by P and p in the same time, by the supposition ; and a
less space than DL would be described in that time by the
motion of P at D continued uniformly (by the first axiom), but
a greater space than (1/ would be described in the same time by
the motion of j^ at I continued uniformly, by the- secoa4
axiom ; so that the velocity of P at D would be to the velocity
ofp at I'm aless ratio t6an that of DL to dl,f>T of E to F. It
appears, therefore, that the velocity of P at D is to the ve*
locity olpstd neither in a greater nor Jess ratio than that of
£ to F, but precisely in this ratio.

29. Case 3. if the motion of P is perpetually retarded, the
motion of p must also be perpetually retarded, because the
spaces described by these points in equal times decrease in the
same proportion. In this case, a less spaee than D6 would
be described by the motion of P at G continued umformly (by
the fourth axiom), and a greater space than dg would be de-
scribed by the motion of p aid continued unifenail;^ ^ the

third

Chap. I. Method ^f fkm&nh ^ I

tiiird axiom)^ in the lime thfit P and p with their lelaided mo-
tions describe DG and % ; so that the velocity of P at O is to
IheTeloeity otp at el in a less ratio than that of DG to dg, or
6f E to F. Tnerefore, if we suppose that the velocity of P at
B is to the velocity ofp at d in a greater ratio than tiliat of E
to F; it feDcms, that the velocity of P at some point betwixt
D and Qj as L, might be to the velocity ofp at d in tlie same
ratio as £ is to F. Bat this is itnpo&»ble. For>.suppo8ing that
DL is to d^ as E is to F^ these spaces will be described by P
and p in the same time ; and a less space than DL would be de*
acribied in that time by the motion of P at Lcontinued uniform*
)y (by the fourth axiom) ; but a greater space than dl would
be described in the same time by the motion ofp at d continu-
ed uniformly, by the third axiom; so that the velocity of P at
li is to the velocity ofp at (/ in a less ratio than that of DL to
d/, or of E to F. It is impossible, therefore, that the velocity
of P at D can be to the velocity of p at </ in a geater ratio
than that of E to. F. Nor can that ratio be less than the ratio
of £ to F. For in the same time that P andp with dieir re-
tarded motions describe DG and d^, a greaVer space than DG
would be described by the motion of P at D continued uni-
formly (by the third axiom)^ and a less space than dg would be
described by the motion ofp at g continued unifoimlyi by the
fourth axiom : so that the velocity of P at D is to the velocity .
pf p at g in a greater ratio than DG is to dg, or E is to F.
Therefore, if the velocity of P at D was to the velocity ofp
Ht d in a less ratio than that of E to F, it might be to the ve-
locity ofp at some point as / betwixt d and g in the same ra*
tio as £ is to F. But this is impossible. For, supposiBg tliat
PL is to ^/ as E is to F, the spaces DL and d/ will be describ-
ed by P andp witli their retarded motions in the same time )
land a greieiter space than DL would be described in that time
by the motion of P at D continued uniformly (by the third ax-
iom), but a less space than dl would be described in the same
time by the motion ofp at I continued uniformly, by the fourth
axiom : so that the velocity of P at D is to the velocity ofp a|
/ in a greater ratio than that of DL to d/, or of E to F. It
appeQrs> therefore, that the velocity of P at D is to the velo*

£4 city

7£ The Grounds of the . Book L

city of p 9Xd ia a rat^io that is neither greater nor less than
that of £ to F^ but is precisely the same ratio.

SO. This theorem may be also demonsbratedj by considering
the spaces that are described by the points P and p before
they come to D and d, whether their motions be supposed to
be continued after that term or not. By joining these cases
together^ the demonstration becomes general; and the sameob-
servation we made in the 2dd article is to be applied here.

^ f

LEMMA I.

31. If Abe to B in a greater ratio than EistoT, and C be to
Din a greater ratio than E is to F; then shall the sum of the
antecedents A andC be to the sum of the consequents B and
D ina gretUer ratio than that ofEtoF.

^ For let G be to B as E is to F, and H to D as E to F;
then the sum of G and H shall be to the sum of B and D as
E is to F. But A is greater than G^ because A is to B in a
greater ratio than G is to B ; and C is greater than H, be-
cause C is to D in a greater ratio than H is to D. Therefore,
the sum of A and C is greater than the sum of G and H ; and,
consequently, the sum of A and C is to the sum of B and Din
a greater ratio than E is to F. In the same manner it appears,
in general, that if there be any number of ratios, each greater
^han the ratio of E to F, the sum of all the antecedents shall
be to the sum of all the consequents in a greater ratio than
that of E to F.

32. It appears, in the same manner, that if A be to B in a
less ratio than E is to F, and C be also to D in a less ratio than
E is to F : then the sum of A and C shall be to the sum of B
and D in a less ratio than that of E to F: and, in general, if
there be any number of ratios, each less than that of E to F,
then the sum of all the antecedents shall be to the sum of all
the consequents in a less ratio than that of E to F.

THEO-

Chap. I. Method of Fbmons. 73

THEOREM VL

53. Whtn the vcldcities of ant/ motions are always to each other
in an invariable ratio, the spaces described by them in the
' same time arc always in the same ratio.

' Suppose that the velocity of the point P is always to the ve-
locity of j? in the invariable ratio of E to F ; let DG and dg
be any spaces described by these points in the same time HV.
Then shall DG be to ifg as E is to F. This is the converse of
the preceding theorem, and may be demonstrated by it and the
fourth theorem : but it may be also demonstrated immediately
from the axioms in the following manner. In the first place> if
the motions of P and p are uniform, it is evident that the
spaces describes by them in the same time are in the same
proportion as the velocities of the motions ; and therefore, in
this case, DG is to Jg as £ is to F.

34. Case 2. Suppose the motions of P and p to be perpetual-
ly accelerated ; and> if the ratio of DG to c2g be greater than that
of E to F, let DK

betodgasEisto AP DL MKN G

F; and DK shall

be less than DG. ^ ^ j/^z.*. -.

•r 1 TTTT a p a I m k n s

Let the time HV — ^- ^

be divided by a H R S Q V '

ONitinual bisecti- ' """*•

on into the equal £ p

parts HR, RS,

SQ, QV, till the time QV be less than that in which P describes
KG. Let DL, LM, I^N, NG be the spaces described by P, and
dlj Im, mn, ng be the spaces described by j?, in these equal times
HR, RS, SQ, QV. Then shall NG be less than KG, because
it is described by P in a less time than KG; and the velocities of
P at L, M, N, and G, will be to the velocities of^ at /, m, n, and
g respectively, ais E is to F, by the supposition. It follows, from
the second ai^iom, that M N is less than the space which would
be described, in the time QV;by the motion of Pat N continued

uni-

f4 The Grounds o/* tie iook L*

uoiformly ; and^ by the first axiom^ ng is greater than the space
which would be described in the same time by the motion ofpat
n continued uniformly. Therefore MNis toiig ii^a less ratio than
the vdocity of P at N is to the velocity ofp at » ; that is, in a less
ratio than £ is to F. In the same manner, IM is to mn, DL U>lm,
and, consequently (by art. 32), BN to Ig in a less ratio than that
of £ to F» Therefore DN is surely to dg in a less ratio than £ is
to F, or DK to dg ; and, consequently, DN is less than DK. But
BN is greater than DK, because NG is kss than KG ; and these
are contradictory. Therefore PG is to ^g in a ratio thai is not
greater than that of £ to F. If it be said^ that DG is to d^ m
a less ratio than that of £ to F ; kt DG be to cfi: as £ is td Ft
and dk will be less than dg. Suppose the time H V to be di^

vided into the e^
A P D L M KN G qua! parts HR,

~ "^ RS, SQ, QV,

p d I m kn s till QV be less

than the time m
H R S Q y T^hicbpdescribea

t J 1 ■■■! II ' '1 riM Ml rS »*■ M I I _ __ -

Ag.LetDI^LM,
E— — F— *^ MN,NOb«th«

s^ces doficiibe^
by P, and dl, im, ntn, ng the spaces described by |>, in the equat
timet HR, RS, Sa QV. 'Hien, si«i» NG ie greater than lh#
q)ace which would be described, in the time QV^, b^ the mo^
tion of P at N continued uniformly (by the first asfiom) \ ^oA
mn is less than the space which would be described, in tlMTaama
time, by the motion ofp at n continued uniformly (b^ the se^
cond axiom) ; it follows, that NG is to mn in a greats ratio
than that of the velocity of P at N to the velocity of p at i^
which (by the supposition) is the ratio of £ to F* In the same
manner, MN is to Im, LM to dl, and therefore lO to dn (bj
art. 3 1 ) in a greater ratio than that of £ to F. Therefore DG
is surely in a greater ratio to du, than that of £ to F, or that
of DG to dft ; and, consequently, dn is less than dk. But the
time QV, in whichp describes 9ig, was supposed to be less thaii
the time in which p describes kg : therefore ng is less than kg^
and dn is greater than dk ; and these being contradictory, it

fol.

Ghap.L MeiiodofFhxiouM. 75

ftaUmi that the ratio al DG to dgh opl less lh«ii ttkut oC
£ to F. It appears^ tberefoie^ that DG isto%af£istoF«
SS. Ca$e S. Suj^xise the motioiia of P aiidj^ to be perpetai^jr
retairded ; and if D6 he to i% in a greater ratio than that of
BteF.ktKQbetoi^asEistoF, andKGwiUbeloMthait
BO. liet the time HV bedirided into equal parts, HR^IIS,
SQ, QV^ tiU HR become kss than the timeki whidiP de-
scribes DK ; let DL, LM, MN, NG be described by the^nunt
P, and dl, Im, mn, ng be described by the point jp, in these
equal times. Then, because DL is described by P in a lei^
time than DK^ DL is less than DK, and 1/3 is greater tha^
KG. But LG must be supposed to be less than KG : for ifi
easily appears^ from die third and fonrtbazioaBS^ that NG is t«^
mn in a kss ratio than thai of the vdooity of P at K to thcirelo-
city €ffp at n, or that of £ to F; and^ ki the same mamier,
it appears, that MN is to im, LM to di, and consecjuently (by
art. 32) LG to dn, in aless ratio than £ is to F ; so that I/i
is surety to ^ in a kss ratio than £ to F,«v KO to i^ : from
which it IcdloWB that lOiolewtlian KG. IfthesatioofDGto
%befttdtobe

kssthanthatof A P D LK M N O

B to F, let D6

be to A:; (less ^ P ^ I k m n g

than djg) as £ is

toF. Suppose, H R S Q V

as before, the

time HV to be E F -

Stthdivided into

the equal parts HR, RS, SQ, QV, till HR becmae less
than the tkne in which p describes dk ; and the spaces DG and
dg being subdivided as formerly, dt shall be less than dk, and
tg greater than kg. But Ig must be less than kg. For it appears,
fiom the fourth axiom, that DL is greater than the space which
would be described, in the time HR, by the motion of P at L
continued uniformly; and Im is less than the space which would
he described, in the same time, by the motion ofp at / conti-
nued unifevmlyiby the third axiom : so that DL is to ^ in a
grealer ratio thaii ttiat of the veleeily of P al L to the velo^

city

76 The Grounds of the Book I.

city of /I at I, or that of E to F. In the same manner^ LM b
to nw^'MN is to f^y and consequently (by art. 31) DN to Ig,
hi a greater ratio than E is to F. Therefore D6 is to ^ u a
greater ratio than £ is to F^ or D6 to kg ; and^ consequently^
1gi% less than kg. Thus it appears^ that when the velocities of
tyhe points P and p are always to each otlier in any invariable
ratio, the spaces described by them in the same time are al*
ways in the same ratio.

THEOREM VII.

«
•

56* When the space described by a motion is always equal to
the sum of the spaces described in the same time by any other
motioms, the velocity of the first motion is always equal to
. ike sum of the velocities of the other motions.

Let the three points P, p, and Q move in the lines AV, au
and EF, and describe the spaces D6, dg, and IH in the s«^me
lime ; and let IH be always equal to the sum of DG and dg :
then diair the velocity of Q be always equaj to the sum of the.
velocities of P and p. If the motions of the points P andp
are both uniform^ then equal spaces being,described by them in
any equal times, the spaces described by Q in equal times will
be also equal, and its motion will be uniform ; and the velo-
city of Q will be to the sum of the velocities of P and/7 as IK
is to the sum of DG and dg, and therefore in a ratio of equality.
37. If the motion of Pbe continually accelerated, and the
motion ofp be. either uniform or accelerated, the motion of Q

is also conti-
A P B D L G V nually acce-

lerated (by
a p b d I g u art. 14) ; and

ifthesumof
E Q ifc I K H F the veloci-

tyofPatD,
added to the velocity otp at d, was greater than the velocity
of Q at I, this sum might be supposed' equal i% the velocity of

Qat

Cha^. I. Method of Flttxions^ rt

Q at some subsequent term^ as when it comes to K. Zjet P and
p desaibe DL and dl in the same time that Q describes IK ;.
and it follows, from the last article, that the point Q» with tts
motion at K continued uniformly, would in this time describe
a space equal to the sum of the spaces that would be described,
in the same time, by the points P andp, with their mo|;ions al
J) and d continued uniformly, that is, a space less than the sum
of BL and dl (by the first axiom) ; and therefore less than I^.
But the point Q, with its motion at K continued unifiormly,
would describe a greater space than IK in that time, by the se-
cond axiom ; and these being contradictory, it follows, that
the velocity of Q at I is not less than the sum of the velocities
of P at D and /I at d. If the velocity of Q at I was great-
er tlian the sum of these velocities, then its velocity at some
preceding term, as when it arrived at i, might be equal to that
sum ; and supposing BD, bd, and £1 to be described by P,p, and
Q in the same time, the point Q, with its motion at k conti*
nued uniformly, might describe a space greater than the sum of
BD and bd (or kl) in the time it describes AI with its accelerat-
ed motion, against the first axiom. This theorem is demon-
strated in the same manner from the third and fourth axioms^
when the motion of P is perpetually retarded, and the motion
ofp is either uniform or retarded.

39. When the motion of P is continually accelerated, but
the motion of ^ retarded, the motion of Q may be accelerated
uniform, or retarded* Suppose, first, that the motion of Q is
accelerated ; and if the velocity of Q at I was greater than
the velocity of P at D added to the velocity ofp at d, it might
be supposed equal to the velocity ofp at d added to the veloci-
ty of P at some subsequent term, as when it comes to L. But
this is impossible. For, supposing that this could be ad-
mitted, and that dl and IK are described hyp and Q in the
time P describes DL, the point P, with its motion at L
continued uniformly, would describe a greater space than DL
in that time (by the second axiom) ; and the point p with its
motion at d continued uniformly, would describe a greater space
than dl in the same time (by the third axipm) ; so that the
point Q, with its motion at I continued uniformly, would de^

scribe

78 The Groimd$ oftU &ook !«

leribe ft greater s^)Bw» than the sum of DL and £& (or IK) in thai
time. But^ because the motion (^ Q is supposed to be ooatiniH^
hf accelerated^ it follows^ from the first axiom^ that the point %
with its motion at I continued uniformly, would describe a less
^nace than IK in that time ; and these being contradictoiy, it ap-
pears, that the velocitj of Q at I is not equal to the vdocity of
Pat Ladded to the velocity of jp atd^and cannot exceed the sum
-of the velocity of P at D added to the velocity of p at d. IS the
.velocity of Q at I was leas than tliis sum, it might be equal to
the velocity of ji aft d added to the velocity of P at some pre*
ceding term of the time, as when it came to B. But this is
impossible. For, if it could be admitted, then, supposing bi
and AI to be described by p and Q while P describes BD, the
point P, with its motion at B continued uniformly, would de-
aoribe a less space than BD (by the first axiom) in that time ;
and the point j9, with its motion at d continued uniformly, would
describe a less space than bd (by the fourth axiom) in tiie same
time : so that the point Q, with its motion at I continued uni«
formly, would describe in this time a space less than the sum of
BD and hd, or k\. But the motion of Q being continually ac*
/ederated, the point Q, with its motion at I continued uni-
formly, would describe a greater space than k\ in that time, by
the second axiom : and these being contradictorjr, it is evident
that the velocity of Q at I is not equal to the velocity of P at
B added to the velocity of p at dy and cannot be less than the
velocity of P at D added to the velocity of /i at d. If the mo-
tion of Q be uniform, while the motion of Pis accelerated and
the motion of /i is retarded, it is demonstrated, in the same
manner, that the velocity of Q at I is equal to the velocity of
P at D added to the velocity ofp at d.

59. If the velocity of Q be continually retarded (the rest
temaining as in the last article), and its velocity at I was great-
er than the velocity of P at D added to the velocity of p at rf,
it might be equal to the velocity of p at some preceding term
of the time, as when it came to b, added to the velocity of P at
D. But this is impossible. For, supposing that this is ad-
mitted, and that P and Q describe BD and kl in the time p de-
fcribes frrf, the point p, with its motion at b continued uniform-

Chap. I. M^iM^lJmkmt. 79

ly^ would describe in that time a space greater thaa bd (by the
third axiom); the point P^ with its motion at D continued
uniformlyj would describe a greater qpace than BD in the same
time (by the second axiom) : and therefore the point Q, with
its mptioQ at I continued uniformly^ would describe iu this time
«space greater than the sum of BD and bd, or kl. 'B«t the
motion of Q being continually retarded^ it would describe a less
space than kl

with its mo- A P B D L G V

tion at I con-'

tinued uni- a p h d I g ^

formly in that

time by the E Q A I K 11 F

fourth axiom:

and these are contradictory. If the velocity of Q at I waa
less than the velocity of P at D added to the velocity of j9 at
d, it might be supposed equal to the velocity of P at D add*
ed to the velocity of p at some subsequent term of the time^
as when it comes to I ; and (by the third and fourth axioms)
it might describe, with its motion at I continued untfbrmly, a
less space than the sum of DL and dl, or IK, in the same tiiM
that by its retarded motion it describes IK; but it would de-
scribe, with the same uniform motion, in tlie same time, %
greater space than IK, by the third axiom ; and these are con-
tradictory. Therefore the velocity of »Q at I is equal to the
velocity of P at D added to the velocity of j? at d.

40. If the space described by Q be always equal to the sum
of the spaces described in thesame time by threeormore points^
it is easy, from what has been shown, to extend the demon-
itration to these cases, by substituting always one point in
place of two : and it appears, in general, that the velocity of
Q at any term of the time is equal to the sum of the veloci-
ties of all the other points at the same term.

4 1 . It follows, from what has been demonstrated, that when *
the space described by any point p is always equal to the dif-
fei'ence of the spaces described in the same time by the points
Q and P, then the velocity of j7 b always equal to the differ-
ence of the velocities of these points.

. THEa

t*

^ . •>

SO The Gtcmds (tf the • Bode L

THEOREM, VIII.

42. When the velocity of a motion is always equal to the sum
of the velocities of any other motions, the space described by
it is ahpays equal to the sum of the spaces desQribed in the
same time by these other motions.

Let the velocity of the point Q be always equal to the sum
of the velocities of the points P and p ; and let IH, DG, and d^
be any spaces described by Q, P, andp in the same time : then
shall IH be equal to the sum of DG and dg. This theorem may
be demonstrated immediatelyfromlheaxioms^inthesameman-
Her as the fourth and sixth ; but more briefly thus : Suppose the

point q to describe^
A P D G y upontfie linee/^a

spac^ ih always
^ p ^ jg u equal to the sum of

DGand rfg ,then(by
E Q r H F the seventh theo-

rem) the velocity of
£ ^ i h f q must be always

equal to. the sum
of the velocities of P andp, and therefore is equal to the velo-
city of Q. From which it follows, by the fourth theorem, that
IH is always equal to ih ; and therefore IH is equal to tlie
sum of DG and rfof.

43. It follows, from this theorem, that when the velocity of
any point p is always equal to the difference of the velocities
of two other points Q and P, the space dg described by p.
is always equal to the difference betwixt IH and DG, which
are described in the same time by the points Q and P.

44. In demonstrating these theorems, we have supposed that
every motion is either uniform, continually accelerated, or con-
tinually retarded ; or that the time may be distinguished into
parts, during each of which the motion is reducible to one or

other

•

H R S Q V

E

F L M N K Z

CSiap. r. Method of tluxwM. 8f

other of those kmds. For though the velocity of a motion may, *
at certain terms of the time, be increased or diminished at once
by a given or assignable quantity, it is impossible that a mo-
tion can be increased ordiminished in this mannerpeipetually,
or at every term of the time. If such a motion could be suppos-
ed, its velocity would exceed all conceivable velocities in the
least tjme that could be assigned. This, if it seem to need a
proof, n^ay be demonstrated in the following manner : Let the
point P describe any space DG, upon the line Aa, in the time
H V ; and let its veloci-
ty at G be to ita veloci- A P . D G a

ty atDin any assignable
ratio, as in that of EK
Id BR The velocity of
P may be supposed, at
certain termsof the time,
to be increased at once, by a quantity that may be in the same
latio to the velocity of P at D, as a given line Zis to EF; but
it cannot be supposed to be mcreaaed in every assignable part of
the time by this quantity, how minute soever it may be. For^
supposing that this could be admitted, let FK be divided by a
continual bisection into equal parts, FL, LM, MN, NK, till
each of these parts become less than Z; and let HV, which re-
presents the time, be divided into the same number of equal
parts, HR, RS, SQ, QV. Then, by what we have supposed,
the velocity of P must be increased, in the time HR, by a
quantity that is to the velocity of P at D as Z is to EF ; and
Uierefore since Z is greater than FL, the velocity of P at the
end of the time HR, is to its velocity at the beginning of that
time in a greater ratio than EL is to EF. In like manner, the
velocity of P being increased in each of the subsequent times
RS, SQj QV, by the quantity represented by Z, which is gi^t-
er than LM, MN, or NK ; it follows, that the velocity of P at
the end of the time HV (that is, when it is supposed to come
to G) is to the velocity it had at the beginning of that time,
when it was at D, in a greater ratio than EK is to EF : but the
ratio of these velocities was supposed tobe the same as that of
EK to EF ; and these are contradiotory. It appears, in the
VOL. L F same

m . Tk Grounds qf the 9ookl^

BfUQe vMSkWif thftt di^ TclocHy of P cannot be supposed to de*
oreasej «o lu^ to b^ diminUbed by a given quaatity^ how minute
soeveo ii^ea^nted by Z, at every tenn of the tim^or in every
assignable part of it.

45. When a motion i» accelerated or retarded in a oontoiue^
mamteTi it is evidentj that» from any given term^ a part of the
time may be taken 8o ami^ll^ that the difference of the v^cities
^ the beginning and end of this time mi^r be less than the dif-
fimnce of any given ungual vdodties ; and that the ratio of
those velocities approaches to a ratio of equality, by diminish-
ing the time, so that it may come nearer to that ratio than
any given ratio of inequality. It follows from thisj that if BB^
DG, and 6K be described by such a motion in the equal timea
HR, RS, SQ ; then by diminishing the tim^ HQ, the. ratio
of QK to BD will approach nearer to a ratio of equality thM
any ratio of inequality that can be assigned. FoTj if tfw.Bto-
tmibe aeceteratedj BD must be graaler than the ippace which

wonld be described in the
A P B D G K ^ tiwie HR by the moticmat

B continued uniformly (by
H R S Q tbe first axiom); and GK
' is leas than the space which

would be de9cribed> in the same time^ by the motion at K con-
tiuued uniformly (by the second axiom) : so that GK is great-
er than "BD, but in a less ratio than the velocity at K is greala:
than the velocity at B ; and therefore in a ratio, that, by di«
minishing the time HQ, may approach nearer to a ratio of
equality than any assignable ratio of inequality. If the motion
of P be retatded in a continued manner, it appears, from the
third and fourthaxioms, that the ratio of BD to GK is less than
the ratio of the velocity atB to the velocity at K; so that, by di-
minishing the lime HQ» the ratio of BD to GK may approach
nearer to a ratio of equality than any given ratio of ineq^iality .

46. It follows, conversely, that if the ratio of GK to BD
approach to a ratio of equality by diminishing the time HQ^
BO as to com^ «earer to it than any giv^en ratio of inequality^
and thus obtain whatever part oS the time be represented by
HQ ; then the motion must be increased or diminished in a

<xm-

r

Chap. t. Method of Ihaiom. Si

eondiHitd BUifmer. Por^ if the motion pcarpetnally increase^
Aeii the vdodty at Q is to the velocity- at D in a les9 ratio
thrna GK is to BD (by the first and aecond axioms) ; and tbefe*
Ibve ia a ratip tiiat (by the 8afqK)Mtioa) may come nearer to
a ratio of equality than any given ratio of inequality : so tibat
die motion mnst be accelerated in a continued m anner . If the
motion ]^rpetiially decrease^ then the velocity at D is to the
velocity at G in a less ratio than BD is to GK (by the third
and foarth axioms) ; and therefore in a ratio l)iat may approach
to « ratio of e^foality nearer than any given ratio of inequali-
tf: «o that themotion is retap&ed in a continued manner. The
motioxiB that am increased or dinliiished in a contknied men-
Ber atvelliQs ^^tinguished from those that at certaih terms are
inoieaeed by « giveil quanti^^ but betwixt those- terms are
eithep imifonti>or<ffe accelerated' or retarded in fL continued
manner. It appears easify^ firom what has been shown^ that^ ilk
the third and fifth theorems, if the motion of P b^ accelerated
<Mr retarded in a continued manner^ the motion of j> is also ac^-
oelerated or retiffded in a eontbitMa manner \ and that, in the
ae^retttb theovem, when the motions of P and p are accelerated
or i^tafded in a continued manner, the motion of Q is atctlG*r
sated or retarded in the some manner.

THEOREM IX.

47* Ifhm « point T deacfihes a lint An rsnth a motion of anjr
AfM, ^mi Moiktfp^Mp de$cfih€8 the same spaces on this Unt
ka, in equaltimes^ but in a donirdr^ order, and with an op-*
posite direetion ; the 'Bel^tiet of these points at afr^ gioe^
term <f tkeline ate eqmL

Suppose that the point P proceeding from A towards^ de-
scribes any ^aeeDL in any time HQ ; and that the point j7,
in moving from a towards A, describes always that space DL
in the same time HQ in which it was described by P ; then
the velocity <rf;> at D mtistbe equal to the vdoCity of P at R

F a When

64 I%0 Grounds of the, Book I«

When the motion of the point P b unifocm, it follows, from
the supposition, that the motion of jp is also uniform, and that
Xheir velocities aie always equal. Let the motion of P be con-»
tinually accelerated ; and it follows, from the supposition, that
the motion of p is continually retarded* If the velocity of the

pointjpatD was

A P D L p a greater than the

*""*'*""**'"*"'*"''~"~"^ """"""""'""'"'"'*'**'*" velocity of P at

H— Q D, it might be

supposed equal to
the velocity of P at some subsequent term of the time, as when
it comes to L. But this is impossible ; for the point P, with
its motion at L continued uniformly, would describe a greatei
space than DL, in the time HQ (by the second axiom); and
the point p, with its motion at D continued uniformly, would
describe a less space than DL, in that time (by the fourth axiom) :
8o that the velocity of |» at D is less than the velocity of P at L.
If the velocity of P at D was greater than the velocity ofp at
X>, it might be supposed equal to the velocity ofp at some term
before^ came to D^ as to its velocity at L. But this also is im-
possible : for the point P, with its motion at D continued uni-
formly, would describe a less space than DL in the time HQ
(by the first axiom) ; and the point jp, with its motion at L
continued uniformly, would describe a greater space than LD
in the same time (by the third axiom) : so that the velocity
ofp at L is greater than the velocity of P at D. It appears,
therefore, that the velocity ofp at any term of the line Aa, as
J), is neither greater nor less than the velocity of P at the same
term, but equal to it. If the motion of P is continually re-
tarded, the motion ofp must be continually accelerated ; and
the demonstration is the same as in the former case.

48. In the following articles, we suppose that w^e the
points P and p describe the line Aa, the points "M and m de-
scribe the line £e, and that EM is determined in any regular
manner from AP ; so that when AP is equal to the absciss of
any .figure, EM is always equal to the correqx)nding ordinate.
Then, if Em be determined in the same manner from Ap, it is
evident, that ni^^en Ap becomes equal to AP, Em becomes equal

to

Chap. t« Method of Umiom. 85

to EM ; and iliat^ according as AP is greater or less than Ap,
EM is greater or les9 than Em. If P and p come together to
any point B, then M and m sliaU come at the same timeto some
point Lupon the line Ee ; and according as DP is greater or
less than I)p^ LM is greater or less than lin. It is also evident,
that the motion of P being uniform^ if the motion of M be ac-
celerated or retarded in a continued manner ; then the motion
ofp being also uniform, the motion of m must be accelerated
or retarded in a continued manner. If tb|js seem to need any
proofs it may be easily deduced from the 45th and 46th article^.

THEOREM X.

49- The motions of the points P andp in the line Aa being uni*

form, kt EM be always determined in any regular manner

from AP, and Em be determined in the same mannerfrom Ap;

then the velocity of the poim M^ at amf term of the line Ee,

shall be to the velocity of the point m, at the same term of that

line, as the velocity of P is to the velocity ofp.

Let the points P andp come together to D ; let M and m
oome at the same time to L : and the velocity of M at L shall
be to the velocity of m at L as the velocity of the uniform
motion of P is to the velocity of the uniform motion ofp. If
the motion of M is uniform^ the motion of ffi is also uniform*
For, if the

point jwde- A P B D G g

scribe the p

spaces FI^

LSin any E M F L , S e

equal times, m

the point p

vrill describeequal spaces BD and DG in these equal times ; and
the motionof Pbeing uniform, it will describe the same spaces
in equal times. But while P describes BB and DG, M de*"
fcribes FIi and LS with its uniform motion ; and therefore FL

F3 ia

gd J%i Ordunds of the feodk t!

is eqnal to lA : and the spaces being equal whidi ttte described
by m in any eqaal times^ its motion must be uniform. Hie ve-
locity of M is to the velocity of P as FL isto BB ; and the
velocity of m is to the velocity ofp m the same ratio. There*
fote the velocity of M is to the velocity of m as the velocity
ofp is to the velocity ofp.

50. If the motion of M is continually accelerated, the mo-
tion of m is also accelerated continually. In this case, if the
velocity of m at I* Was to the velocity of p in a grddter ratio
than the velocity of M at L is to the velocity of J^ ; then the
velocity of m at L might be to the velocity ofp as the velocity
of M at some subsequent term of the line Ee is to the velocity
of P. Suppose the point S to be mxch a term, and let P de-
scribe D6 while M describes LS. Then, because the motion
of the point M is continually accelerated, it would describe a
greater space than LS with its motion at S continued uniform-
ly, in the time P describes DQ (by the second axiom). Prom
which it follows, that the velocity of M at S is to the vdo-
city of P in a greater ratio than US is to DO ; and therefbre
the velocity of m at L is to the velocity of j!> in a ratio that is
also greater than that of LS to DG. But,in the same time that
p describes DG with an uniform motion, the point m describes
liS with an accelerated motion ; and it would describe a less
«pace than LS with its motion at L continued uniformly, in this
time (by the first axiom). Tlierefore the velocity of m at L
is to the velocity of je; in a less ratio than LS is to B6 : and
Ifcese being contradictoiy, it appears, that the ratio of the velo-
city of m at L to the velocity ofp is not greater than the ratio
of the velocity of M at L to the velocity of P. If it was a
less ratio, then the velocity of m at some subsequent term of
the line Ee might be to the velocity ofp as the velocity of M
at L is to the velocity of P/ Suppose the point /to be thai
term ; \etp describe Dg while m describes Lf: aiid since the
point m, with its motion at /continued uniformly, would de-
ficribe a greater space than 1/ while p describes Dg with its
ttiriform motion (by the second axiom) ; it follows, that the
Velocity of m at /is to the velocity ofp in a greater ratio than
V » to jyg: and therefore the velocity of M at^L is to the

velpcity

Chap. T. Method ofFluxiora. 87

velocitj of P in a ratio that is also greater than that of I/to Dg.
But while P describes Dg with an uniform motion^ the point M
describes I/with an accelerated motion; and it would describe a
less space than 1/ with its motion at L continued uniformly^ ia
this time (by the first axiom). Therefore the velocity of M at
L is to the velocity of P in a less ratio than I/is to Dg. And
these being contradictory^ it appears^ that the velocity of m at
L is to the velocity of jp neither in a greater nor in a less ratio
than the velocity of M at L is to the velocity of P. Therefore
the velocity

ofMatLis A P B D G a

to .the velo- p g

city of m at
L as the ve- EM F L S q

■»ii»

locity of P m /

is to the ve-
locity of p. In Uke manner^ this theorem is demonstrated from
the third and fourth axioms, when the motion of M is ooB'-
tinually retarded. In either c««e,it maybe demonstrated from
the same principles^ by considering the spaces described by the
points M and m before they come to L, whether their motions
be continued after that term or*not. An example of the man-
ner how this is done^ was given in the 21st article.

51. It is demonstrated^ in the same manner, that EM being
always determined from AP in any regular manner, and Bm
being determined from J^ in the same manner ; if the motions
of P and M be uniform, and M come to L when P comes to
D, then the velocity of ^ at L shall be to the constant velo-
city of M as the velocity o£p at D is to the constant veloci-
ty of P. If the motion ofp is accelerated continually, the
motion of m is also accelerated continually : and if the velo-
city of m at L was to the velocity of M in a greater ratio
than the velocity ofp at D is to the velocity of P, it might be
supposed that the velocity of m at L is to the velocity of M
as the velocity of jp at g is to the velocity of P ; that is (sup-
posing DG, Dg, LS, I/to be described by the points P, p, M,
m respectively, in the same time), in a greater ratio than Dg is
to DG ^y tiie second axiom), or (because the motions of P

F 4 and

«t J%e JGrounds of the BookL

and M are uniform, andDg is to DG as I/is to IS) in agreat-
er x^tio ^^n I/is to I^. But the velocity of m at L is to the
velocity of M in a less ratio than L is to IS (by the first axiom);
and these being contradictoiy, it appears, that the ratio of the
velocity of m at L to the velocity of M is not greater than th^
yatio of the velocity of |i at D to the velocity of P. In the
same manner, it is shown, that it is not a less ratio ; and there*
fore the velocity of m at I^ is to the velocity of M as the velo-
city of p at P i^ to the velocity of P. When the motions of
p and m are continually retarded, the demonstration is deduced,
in like manner, fi-om the third and fourth axioms.

52. The motion of the point P in the Une A V being uniform,
but the motion ofp variable, let their velocities at D be equal ;
let the time in which p describes AD be equal to the time in
which P describes BD ; and let Dg and DG be also described
by them in any equal times. Th^P^ if the motion ofp be per-
petually ^ccer

A P B P G y l^rated,6Pshan

p b g be always less

than BD (by
the second axiom), because 6D is described byjp with an acce-
lerated motion, and BD is described in an equal time by P with
an uniform motion equal to that which p acquires at D. But
Dg is greater than DG (by the first axiom), because Dg is de-
scribed hyp with an accelerated motion, and DG is described
in an ^ual time by an uniform motion equal to the motion of
p at D. If the motion ofp be perpetually retarded, then (by
the fourth axiom) bD is greater than BD, and (by the third
axiom) Dg is less than DG. •

53. Let the motion of P in the line AV be uniform ; and
the ihotion ofp in the line av be accelerated or retarded con-
tinually (that is, let its velocity increase or decrease fix>m one
degree to another, by passing through all the intermediate de-
grees)^ >vhileP describes any spaces BD and DG ; let/? describe
the spaces bd and dg ; and, the motion of o being first accelerat-
ed, let bd be always less than BD, but dg greater than DG :
then the velocity ofp at d shall be equal to the constant velo-
city of P, For a greater space than dg would be described by

the

Chap. 1 Method of Fhxiofu. 89

thie fointp, with its motion atg continued nnifonnly, in the
same time P describes D6 (by the second axiom) ; and there-
fore the velocity of P is less than the velocity of jp at g. A less
space than bd would be described by the point |»^ with its mo«
tion at b contmued

uniformly, in the A P B D G V

same time P de-

scribes BD (by the (^ P h d g v

first axiom): and

therefore the velocity of P is greater than the velocity of jp at &•
The motion of p is supposed to be accelerated in a continued
manner ; and, therefore, the constant velocity of P being greater
than the velocity oFp at b, but less than the velocity of ji atg, it
must be equal to the velocity of jp at some intermediate term
of the space bg. But it is demonstrated, in the same manner,
that the velocity of P is greater than the velocity of ji at any
term before d, and less than the velocity of jp at any term after
d. Therefore the velocity of P is equal to the velocity of jp at
d. If the motion of p be retarded continually, and bd be al*
ways greater than BD, but dg less than DG i ia ia demonstrat-
ed, in the same manner, from the third and fourth axioms, that
the constant velocity of P is equal to the velocity ofp at d.

54, Let the motion of the point P be also accelerated; but,
the nyotion ofp being more accelerated, let bd be always less
than BD, but dg greater than DG. In this case, the velocity
of P ^t D is less than the velocity ofp atg, because a less space
than DG would be described by the former continued uniform*
ly, and a greater space than dg would be described by the latter
continued uniformly, in the same time (by the^ first and se->
cond axioms). The velocity of P at D is greater than the velo-
city ofp at b, because a greater space than BD would be de-
scribed by the former continued uniformly, and a less space than
M(which is itself less than BD) would be described by the latter
continued uniformly, in the same time (by the same axioms).
Therefore, the motion of the point j!^ being accelerated conti-
nually, the velocity of P at D must be equal to the velocity of
P at some intermediate term of the space bg. But, in the same
manner^ as yrehave showUj^ that it is ^ater than the velocity

of

go The Gratmds of tlic Book L

of jp at 5, and less than the velocity o(p at g, it is demonstrate
ed to be greater than the velocity of p at any term before |»
comes to d, and to be less than the velocity o(p at any term
iaiter it passes^. Therefore the velocity of P at D is equal to
the velocity ofp at d.

' BB. Lettbenotionof the p<Hnt/» be now retarded continn-
ally, and the motion of P be also retarded ; let bd be always
greater than BD, and dg less than DG f which happens when
Uie motion of /> is more retarded than the motion of P). The
velocity of P at D is greater than the velocity ofp at g, be-
cause a greater space than DG would be described by the former
continued uniformly, and a less space than £% would be describ-
ed by the latter continued uniformly, in' the same time (by
the third and fourth axioms). The velocity of P at D is less
than the velocity ofp at 6 (by the same axioms). Therefore,
since the motion of p is

supposed to decrease in a A P B P G V

continued manner, the ve-
locity of P at D must be a p h d g v

equal to the velocity of

p at some intermediate term of the space bg. But, in the same
manner as we have shown, that the velocity of P at D is greater
than the velocity of p at g, but less than the velocity of p at
ft, it is demonstrated that the velocity of P at D is greater, or
less, than the velocity ofp at any term of the space bg, rfonly
excepted. Therefore the velocity of P at D is equal to the
velocity ofp at d,

56. When the motion ofp is retarded continually, and the
motion of P is accelerated continually ; and bd is always great-
er than BD, hut dg less than DG : then the velocity ofp at b
is greater than the velocity of P at B, because a greater space
than fed would be described by the former continued uniformly,
and a less space than BD would be ^escribed by the latter con-
tinued uniformly, in the same time (by the first and third axi-
omsj. Tiie velocity ofpsig is less than the velocity of P
at G, because a less space than dg would be described by the
former continued uniformly, and agreater space than DG would
be described by the latter continued uniformly, in thesflime time

(by

Chap T. Method of Ffuxiom. gi

(by tbe tetxmi and fourth axioms). Therefore^ sSnce the mo-
fioiifl of P and p increase and decrease in a continued manner^
their velocities must be equal at some intermediate term of the
time in fdiich they describe B6 and bg. But^ in the same maa«
ner, as we have shown^ that their velocities are nneqnai at B
and bf and at G andg^ it is demonstrated that their velocities
are miequal at any term of the time^ that only when they come
to D and d excepted. Therefofe, in this case also^ tbe v<
of P at D is equal to the velocity otp at d.

THEOREM XI.

87' The motion of the point P tq>on the line An being uniforM,
And the motion of the point p vpon the same line being acce^
hrated or retarded continually, let their velocities be equal
at D. Tktn, EM being always determined from AP in amf
tegular manner, and 'Eau being determined from Ap in the
tame manner ; tohen P comes to Ji, let M come to L with a
motion that is accelerated or retarded continually, and the
'Bdocity of m at h shall be equal to the velocity of M
ath.

In the first place, let the motion of M be continuaDy
Jsorated^ and the motion of p continually retarded* Let the
points V,p,M, and m describe the ^acesBD, bJ), VL, and/L
respectivdy, in the time TV; and the spaces D6, Dg, IJS, 1/
in the time V/. Then (by art. 52)> bJ) is greater than BD,
and Dg less than DG. From which it fotUows (art. 48)^ that
fL is greater than FL, and I/le$s than LS. In this case, the
motion of m may be uniform^ accelerated, or retarded. Let
it iirst be uniform. By the first axiom, the point M, with its
motion at F continued uniformly, would desccH^e in the time
TV a space less thau FL, which is itself less than/L, the space
described in the same timeby the point jh ; so that the vdoci-
ty of M at F is less thaA the constant velodHy of m. Theie*
ioc^ since the motion of M is accelerated in a continued man*

ner^

9% The Grounds of the Book I.

ner> hy the supposition ; if its motion at L be greater than the
velocity of m, its velocity at some intermediate term of the
space VL, as K^ must be equal to the constant velocity of m*

Let Y,p,
AP B RD QG a M,andi»

p b r
EM F K L «

9 g

N S

describe

the spa*

e ces RD,

m f k

T Z V

t

rD, KU
and kLia
the same

timeZV;

and be-
cause rD exceeds RD (art. 59), kL must be greater than KL.
The point M> with its motion at K continued uniformly, woolel
describe in the time ZV a space less than KL (by the first
axiom), which is less than kL, that is described in the same time
by the point m with its uniform motion. Therefore the velo*
city of M at K is less than the constant velocity^ of nt. But
they were equal ; and these being contradictory, it appears,
that the velocity of M at L is not greater than the constant ve-
locity of m. The velocity of m is less (ban the velocity oC M
at S, because 1/ is described by the former in the time Yt, and
a greater space than IS (which exceeds I/) would be describ-
ed in the saine time by the latter continued unifbnnly, by the
second axiom . Therefore, if the velocity of m be greater than
the velocity of M at L, it must be equal to the velocity of M
at some intermediate term of the space LS, as N. While M
describes LN, let P, p, and m describe DQ, D^, and L« ; and
J)q being less than DQ (by art. 59), Ln is therefore less than
IiN. The point M, with its motion at N continued uniform-
ly, would describe a greater space than LN(by the first axiom),
while m with its uniform motion describes Ln ; and therefore
the velocity of m is less than the velocity of M at N. But they
weree<^ual; and these are contradictory • Therefore the' velo-
city of m is neither greater nor less than the velocity of M at
L, but precisely equal to it. The demonstration proceeds by
the same steps ^hen the velocity of M is accelerated, that of p

retarded^

Ohap. I Method of fltutions. gS

X€tarded^ and the velocity of m i» accelerated/ For it apf^sars^
in the same manner^ that the velocity of m at Lis greater than
the velocity of M at F^ but less than the velocity of M at S ;
so that it must be equal to the velocity of M at some interme-
diate term of the space FS : and it appears^ from the first
and second axioms^ that it cannot be equal to the velocity of M
at any term of the space FS;, but L only.

58. The motion of M being^till accelerated continually,
and the motion of p retarded continually^ let the motion of m
be also retarded. If the velocity of m at L be greater Uian
the velocity of M at L, let the velocity of p in the Une Aa
be greater than the velocity of P in the same ratio*. The con-
stant velocity of p being greater than the velocity of p at D,
and the motion of jp being retarded in a continued manner, the
velocity of

p Ahall be A P » b H g q

equal to the p b g

velocity of

p at some E M m / L / . e

term before mi - a

it comes to

D, as at b. Then, Em being determined from Ap in the same
manner as EM is determined from AP^ the motion of ^n shall
be continually accelerated (art. 48), and the velocity of m at
any term of the line £e, as L^ shall be to the velocity of M at
the same term, as the velocity of p is to the velocity of P (by
the tenth theorem)^ or as the velocity of m at L is to the ve-
locity of M at L. Therefore the velocity of m at L must be
supposed equal to the velocity of m at L. Let j9, p, m, and m
describe the spaces (D, bD^yX and f L^ in the same time ; and
bD being greater than 6D (by the third axiom^ the velocity
of p being equal to the velocity oip at b)y f L must be greater
thanyX. The point m^ with its motion at L continued uni-
formly^ would describe a greater space than fL in that time
(by the second axiom). The point m, with its motion at L
continued uniformly, would describe a less space thanyL
(which is less than fL) in the same time (by the fourth
axiom). Therefore the velocity of m at L is greater than the

velo-

gt !ne GrMfub ofOe Bock X.

yAxiltfotm at tu But these velocititeft wcft tttppesed equal}
and these beii^ coatradictoiy^ it foUows that the yalocity of
m at L it not greater than the velocity of M at L. If the ^-
loctty of mat L be less than the velocify of M at L, let the
"vdocity of p be less than the velocity of P in the same ratio ;;
a]id> since it is less than the velocity of p at D^ let it be equal
to the velocity of |» at some subsequent term of the line Am, a>

• at ^. Em be-

A P j> AD g' a ingdetermin-

p b g ed from Ap

in the same
.......^ E M m / L / $ manner as

^ ^ -^^ J -^"~ EM is deter-
mined from
AP^ the velocity of m at L is to the velocity of M at L atf
the velocity of p is to the velocity of P (by the tenth theo^
rem) ; and therefore must be sapposed ^ qual to the velocily of
in at L. Let j9^ p^ m, and m describe the spaces Dg, J}g, If,
and Ls in the dame time; and Dg being less than J>g (by
the fourth axiom)^ Ls is less than I/. The point m^ with i^
motion at L continued uniformly^ would describe a ^aee less
than Ls in that time (by the first axiom) ; and the point m,
with its motion at L continued unifimrmly^ would describe a
gieater space than I/(which exceeds Lb) in the samet»me(by
the thurd axiom). Therefore the velocity of m at L is less
than the velocity of m at L. But they were supposed equal ;
and these being contradictoi7> it appears that Uie velocity of
m at Lis neither greater nor less than the velocity of M at L^
but precisely equal to it.

59. The motion of M being acoelerated continuaRjr^ as fer-
aedy^ let themoUon ofp be also accelerated coBtiauaRy> and
it is evident that the motion of m is abo accelerated (art* 14).
If the vdocityof matLbenot eqnaltothe velocity -of Mat
h, let it first be greater^ in the same ratio as IV is greater thtfi
IH ; and (R being any point betwixt H and V) let the velo-
city of p be to the velocity of P (<Mr the velocity of p at D)
as IR is to IH; and since it is greater than the velocity ofp
at D, suppose it equal to the velocHy ofp at g. Let p describe

r

Chap. 1. Meikod a^Ftusnom. 95

Dg with its timfonn motion^ in the same time ihal j» deftcribes
J>g, and Dig shall be greater than Dg (by the secimd axiom).
Iiet Em be detennined from Ap in the same manner as EM is
determined from AP ; and^ if m and m describe the spaces Lb'
and I/in that time^ Is shall be greater than I/. The pointi
m> wiUi its motion at L continued uniformly^ would describe «
less space than 1/ in this tkne (by the ftrst axiom). The point
m, with its motion at s continued unifennly, would describe
agreater space than Ls in the same time (by the second axi*'
cm). Therefore the velocity of m at L is less than the vdo-'
city of m at s : but it is greater than the velocity of m at £
(in the same proportion as IV b greater than IR) ; and^ conse*
qneatly^ k must be equal to the velocity of m at some inters'
mediate term of the space Ls^ as o. In the same time that m
describes

Lo^letp^ A P p '& n g a

p, and m p n g

describe

the spa- E M m L o / i

cea MM, m o s

Dn^ and

Lorespec- I H R V •

tiveJy.

The point m, with its motion at L continued uniformly, woiddl
descijbe a space less than Lo in that time (by the first axiom).
The point m, with its motion at o continued uniformly, would
describe a space greater than Lo in the same time (by the se-
cond axiom) ; and, Dn being greater than Dn (because the
velocity of p always exceeds the velocity of p till p come to
g\ ld> is greats than Lo. Therefore the velocity of m-at Ii
is less than the velocity of m at o. But they weie si^posed
equal ; and these being contradictoiy, il appears thait the ve-
locity of w at L is not greater than the velocity of M at L*
In like manner it is demonstrated, that the velocity of m* al L
is not less than the velogity of M at L ; and tbeseibse these
velocities are equal to each other.

^. The other cases of diis theorem, when ^ motioa of 11
is supposed to be letarded continually, aw demoMtiated in th^

same.

96 Tkt Grounds o/ihe Bocrik I.

same manner ; or they may be deduced from those we have de-
scribed by the ninth theorem. When the motions begin or end
at the terms D and h, the same demonstration is applicable ;
since it is sufficient that the motions may be conceived to have
begun before these terms^ or to be continued after them. For
the same reason these demonstrations may be applied, when L
is a tenn where the motion of M ceases to be accelerated,
being afterwards retarded ; or where it ceases to be retailed,
being afterwards accelerated. In like manner, the theorem
may be extended to those cases, when the velocity of M is in-
creased or diminished at L by any finite or assignable quan-
tity, by conceiving the velocity thus augmented or diminished
to have been produced by a continued acceleration or retarda-
tion while M came to L.

61. In general, it follows, from what has been demonstrated,
that when the points P andjp describe- the line Aa with mo-
tions that are either uniform or varied continually ; and, £ ISf
being determined from AP in any regular manner, Em is de-
termined from Ap in the same maoaer : then the velocity of
Ik at any term of die Une Ee is to the velocity of M at the same
term of that line, as the velocity of p, at the corresponding
term of the line Aa, is to the velocity of P at thesame terra*

62. Iii the two following theorems, when we say a ratio is
a limit betwixt two other ratios^ we mean no more, bat that
it is greater than the one, and less than the other.

THEOREM XII.

The velocity of a motion that is accelerated or retarded perpc^
' tually, is, at any term of the time, to the velocity of an uni*
' form motion, in a ratio that is always a limit between the
' ratio of the spaces described by these motions in any equal
• times before that term, and the ratio of the spaces described
by them in any equal times after it.

While the point P describes the line Aa with an uniform mo-
tion, let the point M describe the line Ee with a motion that is

acce-

jj Cliap. 1. Method of Fluxiom. 97

\ accelerated or retarded perpetually. When P comics to D, let

1 M come to L. Let BR and FK be spaces described by the

points P and M in any time before they come to D and L ; and
let QG and NS be spaces described by them in any time after
that term : and the velocity of M at L shall be to the constant
' velocity of P in a ratio that is always a lunit betwixt the^ ratio
of FKto BR and the ratio of NS to QG.

First, let the motion of M be accelerated ; and the point M,
with its motion at N continued uniformly, would describe a
space less than NS, in the same time the point P with its uni-
form motion de*-

scribes QG (by AP B RD QG a

the first axiom).

Therefore the E M F K L N S e

velocity of the *

point M at N is to th« constant velocity of the point P (art. 4)
m a less ratio thaw NS is to QG. But the motion of the point
M being perpetually accelerated, its velocity at L is less than
its velocity at N, and therefore is to the constant velocity of P
in a less ratio than that of NS to QG. By the second aaiom,
the point M, with its motion at K continued uniformly, would
describe a greater space than FK,. in the same time P with its
uniform motion describes the space BR; and therefore the ve-
locity of the point M at K is to the constant velocity of the
point P in a greater ratio than FK is to BR. But the velocity
of the point M at L is greater than its velocity aj: K ; and
therefore is to the constant velocity of P in a greater ratio than
FK is to BR. Thus it appears, that the velocity of the point
M at L is to the constant velocity of P in a ratio that is always
' a limit betwixt the ratio of NS to QG and the ratio of FK to
BR ; being in this case less than the former, and greater than
the latter of those ratios.

63. If the motion of the point M be retarded perpetually,
then (by the third axiom) the point M, with its motion at N
continued uniformly, would describe a greater space than NS,
in the same time P with its uniform motion describes the space
QG ; and therefore the velocity of M at N is to the con-
VOL, I. G stant

98 The Grounds of the Book I.

staat velocity of P in a greater ratio than NS is to QG. But
the motion of M being now retarded^ its velocity at L is great-
er than its velocity at N ; and therefore its velocity at L is to
the velocity of P in a greater ratio than NS is to QG. By
the fourth axiom^ the point M, with its motion at K con*
tinned uniformly, would describe a space less than FK, in
the same time P describes BR with its unifoivn motion ; and

therefoie the ve-
A P B R D Q G a lodty of M at K

is to the constant
E M F K L NS e velocity of Pm a

less ratio than FK
is to BR. But the velocity of M at L is less than its velocity
at K ; and, consequently, the velocity .of M at L is to the
constant velocity of P in a less ratio than FK is to BR. There-
fore the velocity of M at L is to the constant velocity of P
ID. a ratio tliat is always a Kmit betwixt the ratio of NS to QG
and the ratio of FK to BR, being in this case greats than the
former, and less than the latter of those ratios. It is evident^
that in either case the velocity of M at L is to the velocity of
P in a ratio thai; is always a limit betwixt the ratio of KL to
RD, and that of LN to DQ ; for this is only a particular case
of the theorem.

THEOREM Xm.

64* The space described by a motion that is accelerated or rf-
tarded perpetually, is to the space described in the same time
hy an uniform motion, in a ratio that is a limit betwixt the ra-
tio of the velocities of these motions at the beginning of the
time, and their ratio at the end of it.

The points P and M being supposed to describe the spaces DG
and LS in the same time, and the motion of M being accelerat-
ed, as in the 62d article ; then, since the point M, with iu
motion at L continued unifonnly, would describe a less space

than

Qiap;L Method cf Ihaiaiis. 99

than IJSy in the time P describes the qMce I)G with uiaiiifi^^
motion (by the first axiom) ; it foUows^ that the space IS it
to the space D6 in a greater ratio than the Telocity of Mat L
is to the constant velocity of P^ The point M, with its mo*
tion at S continued miifonnly^ woald describe a greater space
than LS^ in the time P deKsibes DG with its .onifarm motion>
(by the second aadcmi) ; and> cdnsequently> the space LS is to
the space D6 in a less ratio than the Teloci^ of M at S is to
tfie constant velocity of P. Therefore the ratio of the space LS
lo the space D6 ib a limit betwixt the ratio of the velocity of
M at L to the constant velocity of P^ and the ratio of the velo*
city of M at S to the velocity of P; being greater than the
former^ and less than the latter of those ratios*

65. Let the motion of the point M be pexpetbally retarded;
as in the 63d article ; and, by the third axiom, the point M
would describe a greater ^ace than LS, with its motion at h
continued uniformly,in the timePdescribesDO; but the point
M, with its motion at S continued uniformly, would tocribe
aless space than IS in the time P describes DG (by te fourth
axiom) ; therefore the space LS is totiiespace DG in aratio
that is less than the ratio 4Df the velocity of M at L to tiie ve^
loctty of P, but greater than the ratio of the velocity of M at
S to the velocity of P.

THEOREM Xm

66. T%e inoHon rf the paint P betng utliyhfm^ but the fnai
the point M continually varied^ let the Telocity off be to the
velocity ofMathasa given line Dgis toJx) let Dg be
alwayzto I/a$the$pace DGdescribedbyfin^u^ timeitto
LS, the epace dewibed by M in the same time. Then, by
diminishk^ thetpaces DG and LS conttnually, qfmay become '
leu them any amgnable magmtudci

Let cr be any small quantity as^iign^ at pleasure ; andtetit
be added to Lc when the motion of M is aocek«alii4»^^ ^^"^ '

G S " dBBtKtd-'

too TJki Gtowds of the Book L

dncteri finm Le ^li£a tfac^ inotioii of M and it is

aiMifeety that^ by diminishing IS, the velocity of M at S ap»
proacheii contimiallj to the velocity of M at L> so that theit
4iftnenca may become equal to. the difference of any two im«
equal velocities that can be assigned, or kss than it, how small
loeverit may be,- Let LS be dhninished till the diffeienoe of
tiloae velocitiesL be to the constant vebdty of P ^ ^^is to I^;

and the velo-
A P B D G g « city of Mat 3

shall be tothe
EM FL S B c f X e velocity of Pas

Lr is to. Dg.
When the motion of M is accelerate, it follows, from the
64th article, that the ratio of LS to DG, or of I/to Dg, is
greater than the ratio of the vdodty of M at L to the veloci-
ty of P> or the ratio of Lc to Dg ; but that the same ratio of
IS to DG, or of I/to Dg, is less than the ratio of the velo-
city, of Bliat S to the velocity of P, or the ratio of Lr to Dg.
Ilieref<ail/is greater than I<> but less than la ; and, conse-
quently, cfh less than ex. When the motion of M is retard^
ed contintially, then (by the 6Sth article) the ratio of IS to

DG, or of I/to

A P B D G ff a Dg,isles$thanthe

ratio of the velo-
EM F L S X f c H e city of M at L to

the velocity of P,
or the ratio of Lc to Dg, but greater than the ratio of the velo-
city of M at S to the velocity of P, or the ratio of Lr to Dg.
Therefore 1/ in this case is less than Ix, but greater than Lr ;
9pd, consequently, cf k less than cx^ In the same manner it is
demonstratqd,, tbuBEt if FL be always to BD as Ls is to Dg, then,
, l^ diminislung the spaces FL, BD, which are described by M
and P before they come to L and D, cs may become less than
any given magnitude. And if FL, LS be spaces described by
the point M in equal times, or in times that are to each other
in any given proportion ; and FL, LS, DG be to each other al-
ways in th^ same proportion as Ls, 1/ and Dg ; then, by di-
minishing the spaces FL and LS continually, ^may become
.••*': * less

• • .

Chap. T. Mithoi of Ffuxibris. tOI

teas than ^ny given magnitude. It appears from what was shown
in the 44tb article^ that any motion must be supposed to be
either uniform^ or varied in a continued manner for some time;
how small soever that time may be ; and therefore this theo«
rem obtains universally.

67* Because ^ the difference betwixt 1/ and Lc decreases so
that it may become less than any given quantity^ how small
^oever^ when DG and LS are diminished continually; it ap-
pears that the ratio of Dg to I/(or of BG to LS) approaches
eontinnally to the ratio of Dg to Lc, so that it may come
nearer to this ratio than the ratio of Dg to any assignable
quantity greater or less than Lc. For this reason^ the ratio of
Lc to Dg is by Sir Isaac Newton called the Limit of the varia-*
ble ratio of I/to Dg, or of LS to D6^. in a more restricted
sense of this term than that in which we made use of it in the
twelfth and thirteenth theorems.

68. When the motion of the point M is continually accele-
rate<i^m L to S, then I/consists always of two parts : the part
Lc isr invariable, and measures the velocity of M at L ; die part
^is variable, and arises from the acceleration of the motkm of
M while it describes LS. This latter part decreases continual^
ly whenDG and LS are diminished, and vanishes with them»
Therefore, when £M is determined from AP by any construc-
tion or equation, and thence the variable 'ratio of LS to DO,
otof I/to the given quantity Dg, is reduced to a rule or ex«
pression, all that is requisite to determine the ratio of Lc to D^
is, to distinguish betwixt Lc the invariable part of I/and th^
variable part cf. And, for this purpose, it is sufficient to sup*
pose DG and LS to decrease, and to find what part of 1/ cou'^
tinually decreases at the same time, and at length vanishes
with LS ; for this part is cf: which being rejected, the re*
mainder Lc is to Dg as the velocity of M at L is to the con«
«tant velocity of P, or as the fluxion of EL is to the fluxion of
AD. When the inotion of M is continually retarded, then 1/
n less than Lc by the difference cf, which decreases and va«
nishes with LS,as before; and this part of the expression of 1/
being discovered and rejected in the same manner, the other
part gives Lc, which is to the given line Dg as the fluxion of
£L b to the fluxion of AD.

G 3 69. It

10> Tke Grounds tjfthe Boole I,

' 69- It u in this concise manner Sir I$aae Newton most
commonly determines £he ratio of the fluxions of quantitiefi.
But we shall treat more fiiUy of his method aftenraids ; and>
smee there have been various objections made against this doc-
trine> we shall demonstrate its principal proportions immedi-
ately from the axioms. By tracing them to such plain prin-
ciples> their evidence may be more easily examined, and ob-.
jecticms against them may either be obviated, or, if any doubt
or difficulty remain, it may appear wherein precisely it lies. It
U worth while to demonstrate the chief propositions of this
method in as dear and complete a manner as possible, if by
Ibis means we can preserve this science from dilutes'. Some
of the preceding theorems are so evident, that they are com-
monly admitted without a proof: hut, because we are deliver-
ing the Elements of this doctrine, and have proposed, in treat-
ing it, to imitate the antient Geometricians (who never in^s
creased the number of their principles without necessity), we
have deduced those theorems from the axioms, that it might
appear how few and plain the principles are which it is neces^
sary for us to assume in demonstrating it. It remains, before
we proceed to enquire into the fluxions of particular quantities,
that we should say something of the higher orders effluxions.
' 70. When a motion is accelerated or retarded continually,
the velocity may be itself considered as a variable or flowing
quantity, and may be represented by a line that increases orde*
creased continually. When a velocity increases uniformly^so
as to acquire equal increments in any equal times, its fluxion is*
measured by theincrementwhich is generated in any given time.
In this case, the velocity is represented by a line that is describ-
ed with an uniform motion ; and its fluxion, by the constant
velocity of the point thatdescribes theline,or by die space which
this point describes in a given time. When a velocity is not ao*
celerated uniformly, but acquires increments in equal times that
continually increase or decrease, then its fluxion at any term of
the time is not measured by the increment which it actually ac-i
quires, but by that which it would have acquired if its acoele*:
ration had been continued uniformly from that term for a given
time. - And, in the same manner, when a motion is retarded

con-

« •

Chap. I. M^kodifthaum. lOS

continually/ the qnantity by which it would he diminiBhed in
a given time^ if its retardatton was continned oniiimnly from
any tenn, measnres its fluxion at that tenn. While the point
M describesthe line Ee, let the point Q describe the line li, so
that IQ may be always equal to the space that would be describ-
ed by the motion of M^ if it was continued unifbnnly for a
given time. Then IQ shall always represent the velocity of
M^andtheve*

locity of the E M e

point Q shall
represent the

flnxion of the I Q t'

velocity of M ; •-'■■■••■•■'^■"■'■"■""•■""•""■^"■""■'^""-■-"■"•■^■■""^

which therefore is measured, at any term of the time^ by the
space which would be described by Q with its motion at that
term continued uniformly for a given time. The velocity of
M is the Auxion of EM ; and therefore the velocity of Q re*
presents the fluxion of the flnxion of EM. Thus, when a flnx-
ion of a quantity is variable^ it may be considered itself as a
fluent, and may have its fluxion, which is called the Second
JluiUm of that quantity. This may also have its fluxion, which
is called the Thirdfluxifm of the first fluent : and we shall show
afterwards, that motions may be easily conceived to vary in.
such a manner as to give ground for admitting second flux*
ions, and those of any higher order.

71. The second fluxions are deduced from the first, in the
same manner and upon the same principles as the first flux-
ions are deduced from their fluents : and th^hefore we shall
subjoin m this place but one theorem concerning them.

LEMMA IL

m

What a motion is accelerated or retarded uniformiy, the space
. described by it is an arithmetical mean betwixt the spaces
. thai would be described in the same time by the motions at
the beginning xind end of that time continued uniformly.

G4 Let

IM nc GrQWvUqfihe Book I.

, Let l]ie.pdixit<1lf de^dib^ tdiW^Hie l& io any given time witli

a motion .that 13 aa^ei:atod or. ii^l^rded uniformly; )et LC and

SH be the spaces tba.t wotdd be .4e(scribed in an equal time by

its motions at Lan4 $ continued wif9n;nly. Then tbedifference

of SH and I^ 9hall be equal to tiie difference of LS and LC.

Let the poini m jmove from S to L^ describing always any

spaces upon SL An times equal to those in which they axe de*

scribed by M^ but in a contrary order (as in art. 47)^ and the

velocity of m at any term of the line LS^ must be equ^l to the

velocity of M when it comes to the same term of that line, by

the ninth theorem. In the same time the point M describesany

dpace Lz, let my describe Sx; and, since the time in which M

describes xS is equal to the time in which m describes it (by

the supposition), it follows, tbat Lz and xS are described by &(

in equal times. Iliierefore, since the motion of M increases or

decreases uni«r

El L ;g R X C $ H_f formly, the

M fn difference of

its velocities
at L and z is equal to the difference of its velocities at x and S ;
and, consequently, is equal to the difference of the velocities^
of m at ^ and S. From which it follows, that the sum of the
velocities of M and m is always equal to the sum of the velo-
citiev of M at L, and of m atS^or to the sum of the velocity of
M at L added to its velocity at S. Therefore, by the eightli
theorem, LS, the space described by M, added to LS the space
described by m in the same time, is equal to the sum of the
spaces LC and SH that would be described in an equal time
by the motions of M at L and S continued uniformly \ Bud,
consequently, the difference of SH and LS is equal to the dif-
ference of IS and LC.

72. Let LR and RS beeqy spaces described by the point M
Wf th a motion that is accelerated or retarded uniformly, in equal
times that immediately succeed after one another; and, in the
same time that M describes the space LS with this motion, it
would describe a space equal to I^ by its motion at R conti-
nued uniformly, l^or the velocity of M at R is an arithmeti-
cal mean betwixt its velocities at Land S, because the amotion

of

Chap. I. M$ihod ^ Buximi^. 106

pf M incieaw or dacaeasw qnifowmly ; aiii4thQiȣ(Meibepouftt
M, with hi AOtiQD Hi R cootinu«d wuforaily, would ^esoribe
a space eq«aliQiia]f tbesumof LCaa4;&H,«i tb^ same iimm
diat it would deaoribe LC with iUaiotioo at h,mS^ with 141
motton at S continaad itmfymif. But, by ^s kmiofiy £4$ ii
oqual to half the au* of LC ttod SU ; au4 ti»e point M 4e«
•eribas LS with iu acoaltoaited Motion lA tba aaiM tvfm tbatit
would describe LC with its motion at L continued uniformly*
Therefore^ in the s^me time that the point M describe^ \S with
amoiioa«ttA£Mn^aooel»atod^itwcHild describe a space equal
to LS with its motion at R continued uniformly.

73. If the motion of the point M begin at ihe term L fronk
nothing; then LC vanishes, and LS is equal to one half of SH ;
thati% when the motion begins fhmi nothings andis accelerat-
ed unifeimly for any time^ the space described by it is ono
half of the qiace described in an equal time by the motion that
is acquired by this acceleration ooQiinuad imifonnly. Thi« U
one of the propositions discovered by Galileui ; and several
othars of this kind may be demonstrated in the same manner-
from the preceding theorems, without having rQcx)nrse to tha
method of indivisibles or of infinitfaimals.

THEOREM XV,

74. Lei the point M describe the Hne Ee with antf variabk
tim; and, in th sam9 time thai it wqM deserve LC» with ii^
motion at L conOmued uHtformly, stj^pose that it would de^
scribe LS, if the ac^eration or retardation if its mqtUm mq$
coHthmed wifformfyfrom thai term. Thm, if the velocity of
Math, or the firsi fluxion of ELj be represented by LQ, thi
secpndflaxion of EL may be measured by SQS.

The fluxion of the velocity of M^ at any term of the tiin^
of its motion, ia measured by the increment which it acquiref^
in a given Un^e^ when its ii^^eleraMon is contiaued uniformly

from

lOS, The Grounds of the Book L

from that term. The acceleration of the motion of M being
supposed to be continued uniformly^ till it describe LS in the
same tinie that it would describe LC with its motion at L oon-
tinned Uniformly ; let SH be. to LC as the velocity which M
would acquire in this manner at S is to its velocity at L : and^
sincfE^ the velocity of M;at L is supposed to be represented by
LCj the velocity which it woald thus acquire at S will be re-
•

£.M F L CSc/ H e

I Q K V t. »

» ii i i • t

presented by SH ; and the increment of its velocity which
would be generated if the acceleratbn of its motion was conti-
nued uniformly from L, in the same time that it would describe
LC with .its motion continued uniformly from h, will be re-
presented by the excess of SH above LC ; which is equal to
2CS> by the last lemma. TbereforeHlre fluxion ^of the veloci-
^ of M at hy or the second fluxion of £L^ is represented by
2CS. W hen the motion of M is retarded, then the decrement
of the velocity of M that would be produced if the retarda-
tion of its motion was continued uniformly from L^in the same
time in which M would describe LC with its motion continued
uniformly from h, is represented by the excess of LC above
SH ; and the second fluxion of £L is represented by that ex-
cess, or by SCS.

75. Or if we suppose, as in the 70th article, that the line IQ
always represents the velocity of M , or the fluxion of EM,
the velocity of the point Q will represent the fluxion of the
velocity of M, or the second fluxbn of EM. Because LC is
supposed to represent the velocity of M at L, let IK be eqnal
to LC, and Q shall come to K when M comes to L. The ve-
locity of Q at K is measured by the space which would be
described by its motion at K continued uniformly, in the same
time thatM, with its motion at L continued uniformly, would
describe LC. Let K V be the space ; and, the motion of Q be-
ix^ supposed uniform while it describes KV, the acceleration or

retarda*

Chap; L Method of Fhx^ions. 1 07

letardatbn of the motion of M i% therefore continued uniform*
ly for that time ; and the velocity of M at S is to its velocity
at Ly as IV is to IK or LC. Therefore the point M would de«
scribe spaccB equal to IK and IV by its motions at L and S con-
tinued uniformly^ in the same time that it describes LS when
the acceleration or retardation of its motion is continued uni-
formly ; and^ by the last lemma, K V is equal to SCS. But
KV represents the velocity of Q at K^ or the second fluxion
of EL ; which is therefore also represented by !2CS. When
the motion of M is retarded^ IQ decreases, the point Q moves
from K towards I^ and the second fluxion of EM is in this case
said to be negative^ being considered as a power that retards the
generating motion^ diminishing continually the first fluxion of
EM. It appears from this theorem, that, as the first fluxion of
a variable quantity, at any term of the time, is measured by the
increment or decrement which would be produced if thegene-*
rating motion was continued uniformly firom that term for a
given time ; so its second fluxion may be measured by twice the
dlfierence betwixt this increment or decrement, and that which
would be produced if the acceleration or retardation of the ge-
nerating motion was continued uniformly from that term fof
the same time.

76. I^t Kv, Ix, and I/be any other spaces that would be
described in the same time by the uniform motion of Q; the
motion of M at L continued uniformly, and the motion of M, if
its acceleration wascontinued uniformly from that term, respec-
tively. Then, the vdocity of M acquired by this last motion
at/shall be to its velocity at L as It? is to IK or LC : and,
the diflerence of those velocities bemg to the velocity of M at
Las ^\a to Lc by the last lemma ; it follows, that ^'n to
Kv as Lc is to LC. But Ko is to KV as Lc is to LC (by the
second theorem) ; and therefore q/'is to CS in the duplicate ra-
tio of Lc to LC. It is also evident, that when LC and LS are
supposed to decrease continuaUy^ the ratio of CS to LC decreas-
es so that it may become less than any assignable ratio. For
the ratio of the velocity of M at S to its velocity at L, or that
pf SH to LC, approaches continually to a ratio of equality, as
the point S approaches to h, and may come nearer to it than

any

K>d The Grounds rf the Book I.

any asiigirable ratio of inequality. But CS is less than the dif*
jGsrence of SH and LC ; and therefore the ratio of CS to LC
may beoome less than any assignable ratio. From this it foi«>
lowsy that when LC^ which represents the first fluxion of £M^
contipoally decreases, then dCS, which represents its second
fluxicm^ decreases so that its ratio to LC may become less than
Iny as^gnable ratio.

^^. Let FL and 1& be spaces described by the point M in
any equal times that succeed after one another ; and let KV be
described by Q in the same time M describes LS. When tike
motion of M is accelerated or retarded uniformly, its vdocity at
Jj, or the first fluxion of EL, may be measured by half the sum
of FLandLS; and the second fluxion of £L may be measured
by the difference of LS and FL. For the point M would de»
scribe, a ^ace equal to half the sum of FL and 1& by its motion
at L continued unifonnly, in the same time that it describes LS
with a OMtkm accelerated or retarded uniformly, by the 7Sd ar«
tide ; and KV, or SCS (which measures the second fluxion of
El when LC measures its first fluxion), is equal to the diflerence
of the spaces 13 and FL. When the motion of M is accele*
nited uniformly, the space \&, which is described by M in a giv*
en time, is equal to LC, that represents the first fluxion of EL,
added to the half of KV, that represents the second fluxion of
£L ; and LS is equal to the difference of LC and one half of
KV^when the motion of Mis retarded uniformly. Inothercases/
when tt^e acederation or retardation ei the motion of M is not
imifonn> but increases or decreases continually while itdescribe»
the space FS» LC is not equal to half the sum of FL and LS ;
bot it follows, firom the 66th and 67th articles, that its ratio to
half their sum approaches continually to a ratio of equality as
its limit, when those spaces are con tinually diminished. And in
the same manner it appears, that, by diminishing LC, the ratio
of KV to the difference of LS and FL, and the ratio of J&
to the sum or difference of LC and one half of KV, confinnltll/
approach to a ratio of equality, so that they may come near^
to it than any assignable ratio of inequality. The ratio of e^
to CS (the diffidences betwixt the spaces described by Mi, and^
those which woold be described in the same tunes by its motio»

con-

Chapv 1 Method of fltanom. tO§

c<mtiiiued4iBiformlj irom L) approaches conyhualt]^ to tbedu*
plicate ratio of Lc to LC, by diminishing thost spaces^ wm»
cases excepted that will be described afterw^acdn Th^re are
theorems analogous to these which relabe to the faigfaer ordeni
9{ fluxions ; but we shall denumstrate then aftsrw A«d% and;
proceed now to enquire into the fluxions of geometricel mag*^
Ailudes. Besides die preceding general tfaeoren»> tbere are
others conoerning the conpMtion and resoktion of motion,
which are sometimes coandered as the grounds of this method s
bnt they may rather ser^e for applying this general doctrine ta
particdtar cases ; and, therefore^ we refer them to another
place.

CHAP, n.

Of, the Fluxions cfpUtne reetiKneal Ftgtatt. '
PROPOSITION I.

78. AHEfluxion.ofaparaildogramofaH momrMk.0iii*
tude u nimaysmtamircd by aparallchgramoftht strnm tUA^
tudt described t^nm tluL right Umc mhkh measures tht^^mxiom
iff tie bate. (Plate 3^ Jig. 16 J

ft

Whik the point P describes the base AO, let the giv^
ri^htline PM> by nMving paraUol to itself, genennte theparal-
lelogram APMF. When Peonies to D^ let PM borne to BE;,
and, if the fluxion of theba^ at thaii:ermiof th^ time be re^
presented by DG, the fluxion of the patfaDelo^am al the* same*
term shall be represented by the iparaUdogronl EG*.

When the motion of P is uiii&cm (that is^ wh^ it describes*
equal qpaoes in any equal tines); the right line PM describes
equal parallelograms in equal tifties (Elem. 36* 1). 'Hierefore
the motion of PM is also uniibna (art. 14)$ aiJd'i'Q the sam^
timethatthe pointP describes DG with ao;uniformmotion> the
right line PM describes the parallelogram EG with an utiiform
motion. There&ie> the flaixiDn of the base AD-beif^ repre«

rented

1

110 Ofihe Flitxidm of Book t

tented Ig^ D6^ the fluxion of the parallelogram AE is repre^
aented by the parallelogram E6^ by art. 1 1 .

79. Sihce P is a point or term of the line PM, which is of a
given or invariable magnitude, and is supposed to move always
parallel to itself j any point in this line moves with the same ve«
locity as P. According as the motion of P is accelerated or
letarded^ the motion of the right line PM is acctierated or re-
tarded ; and, in the same time that the point P would describe
DO with its motion at D continued uniformly, th% right line
t^M would describe the parallelogram EG with its motion con*
tinued uniformly from the same term. If this can be supposed
to need any other proof, it may be demonstrated from the
axioms in the following manner.

80. When the motion of P is continually accelerated, the
motion of PM is also acbelerated continually (Elem. 1.6.
and art. 14). Let the point ;? describe the base with an uniform
motion equal to that of P at D ; lei pm, equal and parallel to
PM, generate the parallelogram ApmiP : and the constant velo-
city of i^m shall be equal to the velocity of PM at the term or
moment when P comes to D. For, while p with its uniform
motion describes any spaces gD andDO>let P with its aoceleraW
ed motion describe the spaces kD and DK ; and since the velo-
ci^ of P at D is equal to the constant velocity ofp, DK is
greater than DG (by ax. 1), and t>g greater than Dk (by ax.
£). Complete the parallelograms B6, Eg, EK, Ek, and £K
dhall be greater than £6, but £i less than £;. By ax. 2, the
right line PM would describe a greater space than EKj with'
its motion atthe term K continued uniformly^ in the same time
P describes DK, or /MM describes £6. Therefore the velociQr
of PM at K is greiLter than the constaiit velocity of pm. By
ax. 1, the right line PM, with its motion at k continued uni«
ibrmly, would describe a less space tiian EX: in the time P and
p describe kD and gD ; and in the same time /mm with a constant *
velocity describes Eg, which is greater than Ek. Therefore -
Ae velocity of PM at £ is less than the constant velocity of '
pm. In tiie same manner it is demonstrated, that the velocity
dfpm is leasthan the velocity ofPMal any term after Ppaases
D> but is greater than the velocity of PM at- any term before

P comes

Chftp. n. ptdne fictiliheal Figures. Ill

P comes to D« Therefore the velocity of pm is equal to thd
velocity of FM at the term or moment ydien P comes to Di
When thetnotion of P is retarded^ it appears in the same man«*
ner, from ax. 3 8c 4^ that the motion of PM at D is equal to
the constant velocity of pm. The uniform velocity ofp, or the
motion of P at D^ being measured by DG^ the motion of pm
is measured by the parallelogram £0 (by art. 78)> which there*
fore measures the motion of PM at D^ or the fluxion of thid
parallelogram AM when AP becomes eqtial to AD« In the
same manner it is demonstrated in general^ that when a given
line, by revolving about a given center or axis, describes any
area ; or when a given surface, by moving parallel to itself, or
by revolving on a given axis, generates a solid ; the motion
with which the area or solid flows at any given term is always
the same, when the velocity of the generating figure at that
term is the same, whatever variation the motion of die gene*
rating figure may be subject to before or after that term«

81 . Let the right lines AO, AV (fig. 17 J, be given in position ;
aad,while the point P describes thebase AO,let the right line PM>
by moving parallel to itself, generate the triangle APM. At
the same time, let the point p describe the right line ao ; and a
given or invariable right linepm, by moviAg parallel to itself^
generate a parallelogram am always equal to the triangle APM«
When P comes to B, let p come to d, PM to D£, and pm to
de : then the velocity with which the base AD flows is the
same as the velocity of the point P at the term or moment when
it comes to D (art. 10), and the velocity with which the tri-
angle ABE flows at that term is the same as the velocity of
the invariable right line jpm when;? comes to d. Therefore^
when the velocity of P is given at any term of the time, to de-*
termine thence the velocity of the right line pm at that term^
is the same as from the fluxion of the base AP to determine the
fluxion of the triangle APM (art. 11); and if the velocity of pm
b given, to determine thence the velocity of P, is the same as
from the fluxion of the triangle APM to determine the fluxion
of the base AP.

82. It is manifest, that when the place of the point P at any
term of the time is given^ the place of the right \ixkepm at that

term

ll€ &f tkt Phtsdcns cf ^kl.

tBrmls ftmittl liy^q^ptying npM the given riglit line <ifapara!<»
klo^ai^ am eqittd td the triangle APM (the i^gle filpo being
afarajrs supposed ieqttdi to- MPO). If the place of ^ is ghren,
andtfaal of P isrequii'ed ; let AO be of such a mi^^itode that
OV paralM to PM may be e^al tadqf orfipi ; and take AP
•saaM propmtibnai betwixt op and AO. Par if £^ be to AP
as AP is to AO, ot as PM ils to OV Cwhich is equal to ^f) ;
this rtsctangle cotttained by ap and of sfaaM be equal to hrif the
neotaagle contained by AP and PM^ and the paraUelogram am
sbsU be eqoal to the triangle APM.

LEMMA lit

S3. fVkinfhd base incnniesuniformlf^f ihe triangle increases with
a imtion that is perp»tna^ atkelem^id ; but when the base

' decreases uniformfy^ the triangk decreases with a motion that
is pcrpttuallyrttaritd*'

< The samethings being supposed as in the tlvblailtartieles; the
Daotion of the point P is the motion wMi which the base flows ;
and the motsonof the given right line j9ff8 is the sameis thatwith
which thetrian^ flows.. Let the right line/>m describethe pa*
nUielograma Adand itt'in any equal times that immediately sac*
eeedaftet onmnother ; aadlec the pointPdeseribe th^tight lines
BD and DG in the same equal thnes: Let BC^ D£, and 6H^
parallel toPM; DMjet APin C> E, and H. Then, because the
i|paces described by jpf0 are supposed to be always equal to the
spitces desdf ibed in the same time by PM^ the parallelbgram be
will be equal to the trapessttrm BDEC^ and the paraHelogifam
dh equal to the trapeaitiin D6HE« Because the motion of P
is uniform, BD is equal' to BG, and the trapezium DGHE i»
gieater than BfiEC in the same proportion as the sunr of DE
and GH is greato: than theaum of DE~and BC^ or the sum
of AG and AD is greater than the sum of AD and AB ; and
the paralielogramifi is greater than hi in the same proportion^
Tiiereibre^ when tha base increases^ uniformly, the spaces be

and

r

Chap. II. plafu rectilineal Figures. 1 1$

and dh described by pm in any equal times that succeed after one
another^perpetually increase; and the motion of jwi is accelerat-
ed perpetually. But Dv len the base decreases^ the point P moves
from O towards A; the right linepn by moving from o towards
a, describes in any equal times that succeed after one another the
parallelograms hd and eb that perpetually decrease ; and its mo-
tion in this case is perpetually retarded. The motion with which
the triangle flows is measured by the motion of pm; and is there-
fore perpetually accelerated when the base increases uniformly^
bat perpetually retarded when the base decreases uniformly.

84. When the triangle APM (fig. 1 S) increases uniformly^ the*
base increases with a motion that is perpetually retarded; but
when the triangle decreases uniformly^ the base decreases with
a motion that is perpetually accelerated. For^ when the tri-
angle APM increases or decreases uniformly, the motion of the
right line|im is uniform, by the supposition. While pm describes
the equal parallelogmms be and dh in any equal times, let the
point P describe the right lines BD and DG ; and, the trape-
zium BDEC being equal to the parallelogram be, and DGHE
equal to the parallelogram dh, the trapezium BDEC is equal
to the tcapezhun DGHE, and BD is greater than DG in the
same proportion as the sum of DE and GH is greater than the
som of DE and BC. Therefore, when the triangle increases
uniformly, or the motion of jpm from a towards o is uniform^
the spaces BD and DG described by the point P in any equal
times perpetually decrease, imd its motion is perpetually re-
tarded. But, when the txvmgle decreases uniformly, or the
motion of pm from o towards a is uniform, the spaces GD and
DB described by P in ai^iy equal times perpetually increase, and
its motion is accelerated perpetually.

86. All the rules for the operations in the direct metliod of
fluxions may be deduced from the two following propositions ;
and there can hardly remain any ground for objectionsagainst it,
whenthese are established in an unexceptionable manner. We
shall therefore demonstrate them at some Icngtb,by themethod
which seems to set the evidence of this doctrine in the clearest
light, and to resolve in the most satisfying manner the diflicul-
ties that have been raised against its ti-uth or ac juracv.

VOL. I. U 'prop.

1 14 Of the Fluxions of Book L

PROP. II.

The sides AD, AE (fig. 17, * 19;, of the triangh ADE
given in position, and the angle ADE being abo given ; in
the same time that the motion with which the base AD Jhws,
continued uniformly, would generate any right line D6, the
motion with which the triangle AVE Jhws, canHnued ^^
formfy would generate the parallelogram £6. Or, iht
fluxion of the base AD being represented by D6, the fluxion
of the triangle ADE is accurately measured by the paralUlo*
gram EG;

While the point P describes the base AO^ and the variable
right line PM by moving parallel to itsdf generates the tri-
angle APM, let the invariable right line/Tm, by moving parallel
to itself along the right line ao, generate the parallelogram mm
always equal to the triangle APM, so that the spaces described
hypm may be always equal to those described in the same time
by PM, as in the preceding articles ; then the motion with
which the base flows, or its fluxion^ shall be always measured
by the velocity of the point P ; and the motion widi which the
triangle APM flows, or its fluxion, shall be always measured
by the velocity of the invariable ]xaepm. When AP bec6mes
equal to AD, let ap become equal to ad (that is, let p come
to d when P comes to D) ; and the right lines PM, pm shall
come to DE and de at the same term of the time. Suppose that,
if the motion of P was continued uniformly from diis term, it
would describe the line D6 in any given time; andthat,ifthemo»
tion of pm was continued uniformly from the same term, it
would describe a space equal to the parallelc^ram dc, in the
same given time : then shall the parallelogram eft be equal ta
the parallelogram EG.

Casel C/?g. 17). When the base increasesuniformly,orthemo-
tion of thepoint P from A towards Oisuniftnm; themgtioaof/wi^

from

Chap. IL - plane ncHUneal Figures. tlS

firom a towards o is a motion perpetually accelerated^ by the last
lemma. Let the point P describe BD and DG in equal times that
succeed immediately after each other ; and let pm describe the
parallelograms bt and dh in the same equal times. Let 6C^ DE^
and 6H parallel to PM meet AV in C^ E^and H; and the pa-
iBllelogram bt shall be equal to the trapezium BDEC^ and dh
eqnid to DGHE, by the supposition. The motion o(pm being
accelerated perpetually^ it follows from ax. 1, that the paral-^
lelogram tk is less than the parallelogram dh ; because the space
dk is described by pm with an accelerated motion^ and ek is the
space that would be described in the same time by pm with its
JDAotion continued uniformly from the banning of that time
without any acoelemtion. By ax. 2., the same parallelogram
€k is greater than be, which was described by pm in an equal
time before its velocity at the term when it comes to d was ac-
qniied.' Therefore the parallelogram tk is less than the trape«
sium DGHE^ but grisater than the trapezium BDEC. It is
evident also^ that the parallelogram EG is less than DGHE^ but
greater than BDEC. I say further^ that the parallelogram ek
is precisely equal to EG. For^ if it is not equal to £G^ it
most be greater or less than it. Let tk first be greater than
EG, and produce BE beyond E to R, till DR be grieater than
DEin the same ratio; and^compIetingtheparallelogramDRLG, .
it shall be to EG as BR is to BE (Elem. 1, 6), or as ^il is to
EG (by the supposition) : and therefore the parallelogram RG
win be equal to tk. Let RL meet CH in N, and NQ paral-
lel to BE meet the base in Q. Suppose that pm would describe
the parallelogram ew, by its motion continued uniformly from
the term when p come to d, in the same time P describes BQ
with its uniform motion. Then^ the spaces described by dny
uniform motion being in the same proportion as the times in
which they are described (by theor. 1. art. l6), the parallelo^-
gram tk, or RG, shall be to theparallelogram ex as B6 is to BQ,
or as RG is to RQ ; and therefore the parallelograms ejp'and
RQ a];e equal. But, while the point P describes BQ^ the right
' line pm describes a space equal to the trapezium BENQ^ by
the supposition ; and, its motion being perpetually accelerated
during this time^ it follows from ax. 1, that BENQis greater

H2

jlfi Of the Fluxibm of Book T.

ih$iSx.ex;^t4pMt which would have been described in the same
time by pm, if its motion had been continued uniformly from
the beginning of that time without any acceleration. Andy
smce DR is greater than D£ in the same proportion as eft is
supposed greater than EG> the parallelogram RQ is greater
than the trapezium DENQ ; and therefore is surely greater than
ex. But RQ was proved equal to ex : and these being contra-
dictory^ it follows^ that the parallelogram ek is not greater than
the parallelogram £6.

86. Let us suppose now that the parallelogram eA is less than
the parallelogram EG ; and^ Dr being supposed less than DE
in the same proportion^ complete the parallelogram rG ; and,
rG being to EG a^ Dr is to DE (Elem. 1. 6)^ or as ek is to
EG; rG must be equal to ek. Let rl produced meet CE in n,
and nq parallel to DE meet the base in q. Suppose that ^nn
would describe the parallelogram ex, by its motion continued
uniformly from the term when p comes to c^^ in a time equal to
that in which P describes jD« Then, by theor. 1^ the pai:alleIo-
gram elc,or rG, shall be to the parallelogram ex as DG is to qD,
ob.as the parallelogram rG is to r^r ; and therefore ex is equal
to rq. ]^t while P describes qj), the right line pm describes
a space equal to the trapezium ^ED, by the supposition ; and,
its motion being perpetually accelerated during this time, it fol-
lows from ax. Q, that the trapezium ^iiED is less than ex, the
space which would be described in an equal time by pm with
the motion continued uniformly which it acquires at the term
when p comes to d. And, the trapezium qhED being greater
than the parallelogram rq (since Dr is less than DE in the same
proportion as ek is supposed less than EG), it follows^ that the
parallelogram ex is greater than the parallelogram rq. But ex
was found equal to rq : and these being contradictory^ it fol*
lows, that the parallelogram ek is not less than the parallelo^
gram EG. Nor is «ft-greater than EG ; and, consequently,
these parallelograms are equal to each other. Therefore, when
P and p come to D and d, if the motion of jpm was continued
uniformly from that term, it would describe a space equal to
the parallelogram EG^ inthesametime thatthepointPdescribe^
DG with its uniform motion : and, the fluxion of the base AD

being

Chap. n. p^ne rectilineal ligures. 117

being represented by J)G, the fluxion of the triangle ADE
(which is measured by the velocity of pm at the term when P
comes to D and/? to d) is represented by the parallelogram EG.
87* Case 2. Let the base decrease imiformly^ or the motion
of P be uniform from O towards A; and the motion of the
right linepn from o towards a shall be perpetually retarded^ by
lemma 3. In this case (the construction and figure being the
same as in the former)^ the space eh is greater than the paral-
lelogram eb, or the trapezium DECB^ by ax. S, and ek is less
than the parallelogram hd, or the trapezium GHED^ by
ax. 4. I say further^ that the parallelogram ek is precisely
equal to the parallelogrfmi EG. For^ ]Sek be not equal to EG,
let it first be greater than EG ; and^ DR being supposed to be
greater than DE in the same proportion, let RL parallel to the
base meet GH in L : and ek will be equal to R6. Then, in
the same mianner as in the 85th article, the parallelogram RG,
or eky is to the parallelogram RQ as the base DG is to DQ : and
it follows from theor. 1, that, in the time P describes QD, pm
would describe a space equal to the parallelogram RQ by its
motion continued uniformly from the term when P comes to
D ; and, by ax. 4^ this spacemust be less than the trapezium
QNED, which is equal to the space that was described by pm
with its retarded motion before p came to d while P described
QD. But the parallelogram RQ is greater than the trapezium
QNED, since DR is greater than D£ in the same proportion as
the parallelogram ek is supposed greater than EG : and these
being contradictory,] it follows, that the parallelogram ek is
not greater than the parallelogram EG. Let ek therefore be
^supposed less than EG, and Dr less than DE in the same pro-
portion ; then, completing the parallelogram tQ, as in the last
article, rQ shall be equal to ek : and since the parallelogram '
rG is to r^ as the base DG is to D;, it follows from theor. 1,
that in the same time P describes J)q, the right line|»n would
describe a space equal to the parallelogram rq, by its mo-
tion continued uniformly from the term when j? comes to d*
But this space (by ax. S) must be greater than the trapezium
DEn^, which is equal to the spase described in the same tim^
time by pm when its motion is perpetually retarded from the

H3 same

118. Of the Flusions of Book I.

same term : and the parallelogram rq is less than the trape-
zium D£»j^, since tir is less tlumD£ia the same proportion as
the parallelogram'^^ is supposed to be less than E6. But these
are contradictory ; and therefore ek is not less than EG. Nor
i^ ek greater tlian EG ; and therefore ek and EG are equal.

.88. Ca$e 3 . Let the triangle APM (fig. 19) increase uniformly^
or the motion otpra from a towards o be uniform ; and the mo-
tion of P from A towards O shall be perpetually retarded, by
art. 84. In thiscase^ if ek be supposed greater than EG, letDR
be greater than D£ in the same proportion; and the parallelo-
gram RG shall be equal to ek. Let RL parallel to the base
meet EH in N, and NQ parallel to D£ meet the base in Q.
Then, since the parallelogram RG is to RQ as DG is to DQ;
and it is supposed, that, in the time pm describes the parallelo-
gram ek (or RG) with its uniform motion, the point P would de-
scribe DG with its motion continued uniformly from the term
when it comes to D : it follows, firom theor. 1^ that the point P
would describe DQi by th^ same motion continued uniformly,
in the timepn describes a space equal to RQ. Therefore the
point P would describe a line less than DQ, by the same motion
continued uniformly, in the timepm describes with its uniform
motion a space equal to the trapezium DENQ, which is less
than RQ. But while pm describes a space equal to DENQ,
the point P describes DQ with a motion perpetually retarded
(by art. 84), and therefore, by ax. 3, it would describe a line
greater than DQ in this time by its motion continued uni-
tbrm]fy irom. the term when it comes to D : and these being
contradictory, jt appears that ek is not greatei^ than EG. Let
eA: be. now less than EG; and, if Dr l)e less thanDE in the
same proportion, ck shall be equal to rG. Let rl parallel to
the base meet CH in », and nq parallel to DE meet the base
in q. Then, since the parallelogram rG is to r^ as the baseDG
is to l)y, it follows, from theor. 1, that, in the same timepn
describes a space equal to rg with its uniform motion, the point
P would describe a line equal to Dy by its motion continued
Vniformly from the term when it comes to D; and therefore P
Vrould describe a greater line than D j, by the same motion con-
tinued uniformly, in the time/}m describes a space equal to the

trapezium

V\£XeXAhi//8y^lJ.

B IX a O

f' d

%

* /

i

4

i .^

Chap. U. plan^ rectitineal Figures. 1 ig

trapftzinm gnED, which is greater than the parallelogram rq.
Bat in this time the point P describes ^D with a retarded mo-
tion (by art. 84X and therefore it would describe a less line
than qD in the same time with the motion that remains when
it comes to D continued uniformly^ by ax. 4. And these be^
ing contradictory, it follows, that the parallelogram ek is not
less than EG. Nor is ek greater than £6 ; and therefore these
parallelograms are equal to each other.

89. Case 4. Let the triangle APM decrease uniformly, or the
motion ofpm from o towards a be uniform ; and the motion
of P from O towards A shall be perpetually accelerated, by
ait. 84. In this case, if ek was equal to any parallelogram RG
greater than EG (the construction being similar to that of the
last article), the point P would describe a line equal to DQ,
by itsmotion continued uniformly froAi the term when it comes
to D^ in the same time pm with its uniform motion describes a
space equal to the parallelogram RQ ; and the point P would
describe a line less than DQ, by the same motion continued
uniformly, in the tame pm describes a space equal to the tra-
pezinm QNED. But, while pm describes a space equal to this
trapezium,'the point P describes QD with an accelerated mo-
tion ; and it would describe a line greater thanQD in this time,
with the motion it acquires when it comes to D continued uni«
formly, by ax. 1. And these being contradictory, it follows
that ek is not greater than EG. If eA was equal to any paral*
lelogram rG less than EG, then the point P would describe D^^
by itsmotion continued uniformly from the term when it comes
to D, in the same time pm describes a space equal to the pa-
rallek^am rq ; and P would despribe a greater line than J)q,
by the same motion continued uniformly, in the time pm de-
scribes a space equal to the trapezium DEn^r, which is greater
than the parallelogram rq. But, while pm describes a space
equal to DEnq by its uniform motion, the point jp describes the
line Dq with an accelerated motion ; and it would describe in
this time a less line than Dq, with its motion continued uni-
formly from the term when it comes to D,by ax. 1. And these
being contradictory, it appears, that the parallelogram ek is
not less than EG. But k is not greater than EG ; and there*

H 4 fore

120 Of theFluxions of ' Book I.

fore these parallelograms are equal to each other^ whenetthe^
the base AD, or the triaogle ADE, increase or decrease uniformly,
go. The last three cases might have been demonstrated from
the first, by art. 47 8c 57* But, for the illustration of this me-
thod, we chose to deduce them immediately from the axioms
and the first theoi^m. When the motion ofpm is accelerated,
and the motion of P accelerated or retarded, the prc^sition
may be demonstrated in the same manner. But all the other
cases of this proposition are briefly deduced from the first, by
the eleventh theorem, thus. In general, let the motion of P,
while it describes AO, be accelerated or retarded at pleasure ;
but let it come to D with a motion that, continued uniformly for
any given time, wouM generate the line DG : and therightline
pm shall come to de with a motion which, if it was continued
Uniformly for the same time, would generate a parallelogram ek
equal to the parallelogram EG (fig. £0). For, suppose QN pa-
rallel to PM to generate the triangle AQN; let the motion
of Qbe unifoim, and equal to that with which P comes to D;
let qn, equal and parallel toji^m, generate the parallelogram £ij^^
always equal to the triangle AQN; and, if the motion of j^n was
continued uniformly from the term when jr comes to d, it would
describe a space equal to the parallelogram EG in the same time
Q describes DG, by the first case (art. 85 & 86). Therefore,
since the velocity of Q is constant, the velocity of ^n increases
or decreases as DE, or as AD increases or decreases ; and, con*
sequently, it is accelerated or retarded in a continued manner.
It is evident, that aq is determined from AQin the same man-
ner as ap is determined from AP. Therefore, since the points
P and Q are supposed to come to D with equal velocities, the
points jp and q shall come to d with equal velocities, by theon
11, and the velocity of pm at that term is equal to the velocity
of ^. From which it follows, that if the motion ofpm was
continued uniformly from the term when^ comes to d, it would
describe a space equal to the parallelogram EG, in the same
time that the point P would describe DG with its motion con-
tinued uniformly from the same term. Therefore, in general,
the motion with which the base AD flows would generate DG,
and the motion with which the triangle AD£ flows, continned

uniformly.

Chap. II. . plane rtctUintal Rgurct. 1£1

uniformly^ would generate a space equal to the paralielogrant
£6, in the same time : sothattheflaxionof thebaseADisre*
presented by T)G, and at the same tim^ the fhudon of the tri«
angle ADE by the parallelogram £0.

91. Corollary I. The motion with which the trian^e flowt
is the same^ whether it increase or decrease with an uniform or
with a variable motion^ when thebase is of the same magnitiide^
and flows with the same motion. For> when AD and DG avo
given^ the parallelogram EG is of a given magnitude*

92. Cor. II. The triangle ADE (fig, 9\), any trapezium
ABEF when the line FE is given in podtion^ and the paral-
lelogram ADEL when the side AL is invariable^ flow: with the
same motion when the motion with which their base AD flows
is given. For all those motions continued nnifcnrmly woul4^
generate the same parallelogram EG ii| the same time.

93. Cor. III. While the base AD by increa»ng uniformlj
acquires the augment DG> the triangle ADE acquires the aug«
ment DGHE. But it is. only the part EG of this augment that
can be said to be generated by the motion with which the tri-
angle ADE flows at the term when P comes to D. The part
EIH is generated in consequence of tbe accelerations of this
motion : for the space EG is all that would be generated by
the motion with which the triangle ADE flows, if that motioii '
was continued uniformly, without any further acceleration^ for
the time in which the base AD acquires the augment DG. If
the motion of /mi was to be accelerated no more after it amves
at dy then we have shown that pin proceeding with an uniform
motion would describe a space equal to £G^ and not to the tra-
pezium DEHG^ in the time P describes DG ; and^ in measnr*
ing this motion of pir, or the fluxion of the triangle ADE,
the part EIH of the increment which the triangle acquires in
this time ought to be rejected. When the base decreases uni-
formly, the triangle decreases with a retarded motion, the pa^
rallelogram EB> or EG, is equal to the space that would bis
generated by this motion, or the motion oipni at d, if it was
continued uniformly, while P describes DB : but a less space
DECB is generated in this time by the retarded motion with
which the triangle flows; and the diflerence ECS arises firom

the

1S3 Qf^^ Fluxions o^ Book I.

the retardation of that motion. In general^ whether the.mo-
tion c^ P be uniform or variable; whatever is generated by the
motion with which the triangle flows more than the parallelo-
gram £6 (in the time P would describe DG by its motion at D),
arises from the acceleration of this motion ; and whatever is ge-
nerated less tj^an £G is owing to its retardation. Those accele-
rations or retardations may observe various laws. They depend
upon the motion of the point P before or after it comes to the
-term D ; and are difiSsrent when the point £ describes different
light lines, But the motion with which the triangle AD£ flows
when P comes to D is not affected by them^ and depends upon
the motion of P at the term when it comes to D and the magni*
tnde of the right line D£ only^ the angle £DG being given.

94. Cor. IV (fig.lS). When the base increases uniformly^ the
triangle increases with a motion thai is uniformly accelerated.
For, if DG be described by P with any uniform motion in a
given time^. DG will be of an invariable magnitude^ and the
parallelogram £G will increase in the same proportion as D£ or
^D^ and therefore will increase uniformly ; so that the veloc^i-
ly ofpnij or of the point p, will be as the time from the begin-
ning of the motion^ supposing the point P to begin its motion
at A. When the base decreases uniformly, the motion with
which the triangle flows is uniformly retarded : for it decreases
in the same proportion as D£ or AD decreases ; and the velo-
city of pm, or of the point p^ decreases in the same proportion
as the time that remains to flow till P come to A. The mo-
tion of pm, or ofp, in the first case^ is similar to that of heavy
bodies descending by the action of influence of an uniform
gravity ; and the motion of pm, if it. was to be. accelerated no
more after it comes to d, would be similar to the motion of a
heavy body^ if^ after a like term^ the action of gravity upon it
was to cease^ or was destroyed by an equal opposite action^ pres-
sure^ or resistance. . And thus it appears^ that we have made no
suppositions^ in demonstrating this proposition^ but such as aie
not only easily conceived (which however is all that is requir*
ed in geometry) and generally admitted^ but are also founded
in nature and conunon observation^ and are illustrated by the
motions thsd; aie most universally known* When the base AD

decreases

Chap. II. plane rectilineal Figures. 1£3

decreases uniformly, the motion o£pm, or of the point p, is si-
milar to the motion of a heavy body rising in a line peq^endi-
cnlar to the horizon against the action of an uniform gravity.
95. Cor. V. It may be worth while to observe, that the doc-
trine of motions that are accelerated or retarded uniformly is
easily demonstrated from the two first cases of this proposition^
without supposing any quantities indivisible or infinitely dimi-
nished. When the motion of P is uniform, AP represents the
time ; the space described hy pm, being always equal to the tri-
angle APM, increases in the duplicate ratio of the time AP
(Elem. 19. 6), and the spacedescribed by the point p increases
in the same proportion. The velocity ofp increases uniformly,
in thesame proportion as AP,the time/rom the beginning of the
motion, by the last corollary. From (fig. 2 1 ) whichit follows,con-
versely, that if the motion of any point, as p, be accelerated
uniformly, or in the same proportion as AP, tiie time from the
beginning of the motion, the space described by it from the
beginning of the motion shall increase in the duplicate ratio of
this time. For, if the velocity of p be equal to the velocity
ofp at any term of the time, their velocities, and consequently
the spaces described by them in the same time (by theor. 4)^
must be always equal. And this is the celebrated theorem dis-
covered by Galileus. Because the parallelogram EG is to
the parallelogram A£ as the base DG is to AD ; it appears, that,
in the time P describes AD, the right line pm would describe a
space equal to the parallelogram AE, if the motion it acquires
at d was continued uniformly. But the parallelogram AE is
double of the triangle AD£, or of the parallelogram ae. There-
fore the point p would describe a line double of ady if the mo-
tion it acquires at <2 was continued uniformly, in the same time
that it describes ad with a motion that is accelerated uniformly :
which is another of his theorems. Because the trapezium BCH6
is double of the parallelogram EG, it follows, that the space
described by a motion that is uniformly accelerated is equal to
the space that would be described in an equal time by the motion
at the middle term of that time continued uniformly ; as was
shown in a different manner in art. 7£. When the base AP de-
creases unifoxmlyj the velocity ofp from towards a decreases

nni-

124 Of the Fluxions of' Book I,

uniformlj^ and may be always represented by AP. The space.
pa that remains to be described by the point/? before its motion
be at an end, decreases in the same ratio as the triangle APM
decreases, or in the duplicate ratio of AP ; that is, in the du-
plicate ratio of the velocity ofp, or of the time that remains to
flow till the end of the motion. Therefore the spaces that may
be described by bodies before their motions be destroyed, when
they are continually retarded by an uniform gravity (or by any
nniform power or resistance that diminishes their velocities
cqaully in any equal times), are in the duplicate ratio of their
velocities.

96. Cor. VI. If the angle APM be a right one, and PM be
double of AP, the triangle APM shall be always equal to a
square described upon AP ; and the motions with which this
square and the triangle APM flows shall be always equal, by
theor. S. Therefore, when P comes to D, the motion witli
which a square upon AD flows, continued uniformly, would
generate a rectangle equal to EG, in the time P describes DG.
But the rectangle EG is equal, in this case, to a rectangle con-
tained by 2AD and DG; and this rectangle represents the flux-
ion of the square, when DG represents the fluxion of AD the
side oTthe square. The parallelogram ad being equal to the
square of AD, the right lines of, AD, and ad are in continued
proportion ; and, the rectangles ek and EG being equal, DG
(which represents the fluxion of AD) is to dk (which represents
the fluxion of ad) as of is to DE, or 2 AD ; and DG is to one
half of dk in the subduplicate ratio of of to ad. Therefore,
when three quantities are in continued proportion, and the first
is invariable, the fluxion of the second term is to the fluxion of
the third as the first term is to twice the second term ; and the
fluxion of the second term is to one half of the fluxion of the
third term in a subduplicate ratio of the first term to the third,

97. Cor. VII. When the base flows uniformly, the velocity
of the motion with which the triangle flows increases in the
same proportion asPM or AP increases, and is equally augment-
ed in any equal times. • Therefore the fluxion of this velocity,
or the second fluxion of the triangle, is constant : and if we
conceive the continual acceleration of this velocity to be the

effect

Chap. II. plane rectilineal Hguru, 125

effect of some power^ the exertion of thid power must be sup^
f>osed uniform. When P comes to D, the motipn with Which
the triangle flows^ continued uniformly^ would generate the
parallelogram EG in the same time P describes D6. When P
comes to G^ the motion with which the triangle then flows,
continued uniformly^ would generate the parallelogram DM in
the same time. The difference of these parallelograms^ or the
parallelogram IN^ represents the second fluxion of the triangle
ADE ; and IN is double of the triangle EIH^ which is the part
of the increment of the triangle that is generated inconsequence
of the uniform acceleration of the motion with which it flows
while P describes DG. It is evident, that if the motion i>fP
be increased or diminished, then the motion .with which the
triangle ADE flows^ or its fluxion, is increased in the same pro*»
portion ; because the parallelogram EG increases in the same
proportion as its base DG. But the trian^e EIH, or the paxsii^
lelogram IN which measures the second fluxion of the trian-
gle^ increases or decreases in the duplicate ratio of DG^,or of
the space EG which measures its flrst fluxion. And, the velo-
city of P remaining, if the time in which DG is supposed to. b6
described be continually diminished, then {IG which measures
the first fluxion of the triangle decreases in the same propor-
tion ; but IN is diminished in the duplicate ratio of this time,
and its ratio to EG may become less than any assignable ratio.
The parallelogram IN (which measures tlie second fluxion of
the triangle ADE) is equal to the difference of the increments
DECB, DGHE which are generated while P describes BD and
DG. The increment DGHE is equal to the parallelogram EG
(which measures the fli*st fluxion of the triangle) added to one
half of IN which measures its second fluxion. Tbete things
agree with the 74th, 75th, 76th, and 77th articles, and illustrate
them. When the base flows uniformly, it is evident that the
triangle has no third fluxion : and what we have said of the
triangle APM is easily applied to the right line op, which
always flows in the same manner, because the rectangle con-
tained by it and the invariable right line o^is always equal
to the triangle, by the supposition.

I

LEMMA

ISfi Cftht Fbmon$ of Book L

LEMMA IV.

ge. Let ihepoinii P a^ Q(fig. 22 4r 25) describe the right Une$

AO^ AV given in position, with motiam that are either imu

fortft, or are always to each other in an invariable ratio ;

rnnd, the parallelogram APRQ being completed, the point

R shall be always found in the same right like.

WhQe the point P describes BD and DG upon the line AO,
iet Q describe KL and LM upon the line AV ; complete the
ptfallelograms ABCK^ ADEL, and AGHM ; let CN parallel
to AO meet DE and GH in S and N ; and let EI parallel to
AO meet GH in I. Becanse the velocities of P and Q are
t0 eaeh other in an inrariable ratio^ it follows from theor. 6,
that BD is to DG as KL is to LM. Therefore BD b to BG
aa KLis to KM^ or CS is to SE as CN is to NH ; and, con*
aeqnently, any three places C, E, and H of the pcnht R being
ih a right line, it follows, that the point R is always found in
the same right line.

PROP. III.

99. The fluxions of the right lines AD and AL (fig. 22 & 23)
being represented by DG and LM, the fluxion of the rectan^
gle AE, contained by AD and AL, is measured accurately
by the sum of the rectangles £lG and EM when these lines
increase or decrease together, but by the diff^erence ofBQ and
EM when one of those lines decreases while the other
increases.

Case 1 . Let the sides AP, AQ (fig. 22), of the rectangle AR
both increase, or both decrease together,by theuniformmotions
of the pointsP andQ. When P comes to D, let Qcome to L,
and R to E; and while P describes DG, let Q describe LM, and
R describe EH : and the point R sh^sdl be always found in the

right

€3iap. n. plane reetilmtal figures. Ijfel7t

right line EH^ by the last kmma. Produce H£^ and let it
meet AD m F before it meets with AL : then^ the rectangle
AR being always equal to the sum of the triangle FPR and
trapezium AFRQ^ the increment or decrement of the rectangle
is always the sum of the increments or decrements of the tri-
angle and trapezium that aro generated in the same time ; and
by theor. 7j the motion with which the rectangle flows is al«
ways equal to the sum of the motions with which that triangle
and trapezium flow. But^ by prop. 9, (art. %5\ if the motions
with which the triangle FDE and trapezium AFEL flow were
continued uniformly, the parallelograms EG and EM would be
generated by them in the same time that the points P and Q
describe D6 and LM with their uniform motions. Therefore,
if the motion with which the rectangle AE flows was coa-«
tinned uniformly^ a space equal to the sum of the parallelo*
grams EG and EM would be generated by it in the same
time ; and, if the fluxions of the sides AD and AL be repre-
3ented by DG and LM, the fluxion of the rectangle AEmust be
represented by the sum of the rectangles EG and EM. The de-
monstration is the same when HE produced meets ALbefore it
meets with AD : only, in this case, EM represents the fluxion
•f the triangle, and EG the fluxion of the trapezium ; and if
HE produced pass through A, the point F coincides with A,
and the rectangle AEis the sum of two triangles, the fluxions
of which are represented by the rectangles EG and EM.

100. Casf 2. Let the side AQ (fig, 9S) decrease while the
side AP increases; and the motions of P and Q being still uni«
form,thepointRshalldescribearightline£H,by the last lemma.
Because the angle MHEis acute, andtheangleHMLright,HE
produced shall meet with AL produced beyond L in some point
f; and it shall meet with AD produced beyond D in some point
F. The rectangle AE in this case is always the excess of the
trapezium ADE/'above the triangle/LE ; and the motion with
which it flows is always equal to the difference*of the motions
with which that trapezium and triangle flow, by art. 4 1 • There-
fore, if the fluxion of the sides AD and AL be represented by
DG and LM, the fluxion of the rectangle AE shall be repre-
sented by the difierence of the rectangles EG and EM.

101. Ca$c

19S Of the Fluxions of Book L

101. Case S. When the .motions of the poiiits P and Q are
variable, but are in aa invariable ratio to each other^ the point
R stiH cl^cribes a right line, by the last lemma ; and the mo-
tion with which the rectangle A£ flows is still measured by*
the sum or difference of the rectangles EG and EM when the
motionsof Pand Q, at the term when they come to D and L, are
measured by DG and LM^byprop. 2. In general, let the motions
of P and Q (fig. 24) be varied at pleasure ; and let them
^me to D and L with motions by which, continued uniformly,
they would describe DG and LM in any given tune. Let the
right line AC constitute with AD an angle tliat is half a right
one ; produce ED and EL till they meet AC in C and K : and
KE, or C£, shall be always equal to the sum of AD and AL.
Upon AD produced let AF be always equal to CE, and FR pa-
rallel to DE meet AC in R ; and the triangle AFR (which is
equal to KEC) shall always exceed the rectangle AE by the
two triangles ADC, ALK. Therefore the motion with which
the rectangle AE increases is equal to the excess of the motion
with which the triangle AFR increases above the sum of the
motions wi th which the triangles ADC, A LK increase. When
the sides AD, AL increase together, take FT equal to the sum
of DG and LM, but equal to their difieience when AL de-
creases while AD increases. Let GQ and TX, parallel to D£,
meet AC in the points Q and X ; let MY parallel to AD, meet
AC in Y; let CP and RZ, parallel to AD, meet GQ and TX
in P and Z ; and let KS, parallel to AL, meet MY in S. By
prop. 2, if the motion with which the triangles AFR, ADC,
ALK, increase were continued uniformly, they would generate
the spaces FZ, DP, LS in the same time tiiat the motions with
which their bases increase, continued uniformly, would gene-
rate the right lines FT, DG, LM. But, when the sides AD,
AL increase together, FT is equal to the sum of DO and LM ;
and FR is equal to KE, or EC: and therefore FZ is equal to
the sum of the rectangles ES and EP. From which sum de-
duce DP and LS, which measure the motions with which the
triangles ADC, ALK increase, and the remainder is the sum of
EG and EM ; which therefore measures the motion with which
the rectangle AE flows. But if AL decrease while AD in-
creases.

Chap. IX. plime rtetilkkat Figures. \%^

creases, ibeii^ FF beiag eqikai to Ae differeift^e on)G add i^

SD9f<ciBciecBBD^9B ji/ aLt^ruiDe eQCMttuyxuc CEmeiezice oi uieFedaiv*

gleb £P aod £S; and, because the tnangle A LK decreasea^the BMK
' tba witkiAichihe simi of the taiangles ADG^ ALK hltr^as^

be measui e d by the diffeience betmzt DP and l&. From F2^
6r the diflkrencie betwixt £P and ES, sobduct the difference be*^
twist DP and LS^ .and the remainder will be equal to the differ-
enee betmxt EG atnd KM ; which therefoite measures the mo^
tionrvKth which the rectangle A£ flows when D6 and LM meftt
sure the moticms widi which its side» AD and Aliflow. If DG
be ecpial to LM, the pvoralletogram EP is eqnnd to ES ; the line
FT, and the motion- with which the triacfgk AFR flow^, vK*
nishr; themotionwiAwbichthe rectangle AEincreasesisequal
to the motion' with which &cf Sum of di^ triangles ADC, ALK
decreases, amd therefore is meaisored by the difference betwixt
IS and DP ; which is equal to the difference betwixt EG and
£M, because ESrthe sum- of EM and LS is equal to EP the sum
ofEGandDP. If LM be greater than DG, theti the triangle
KEC (or AFR) decreases, a^d FT ikmst be tahen from F to-
wards A : but the motion widi which AE idcreases will still be
measured by the ekcess of EG above EM. When EG is equsi
to EM, then this iliotfon vanishes ; and, when EG' i^ less than
EM, the rectangle AE decreases by a motion thatis still niesl-
sored by their diflerenoe. Thus it appears) thiW, whether the
sides AB and AL ffow with noiform or variaUe motions^ and
the fpoint E describe a right Icne or ikot, when the fknions of.
the ftdes AD and AL ore measotfed by DG and LM> the dusS
ionof the rectangle AB is measured by die sum ofdifibeiiec
of the rectan^es EG and ]EM, their sum wbeirthesides in*
crease or decrease together, their di ffe rco ce i when one sidedeA
cttmf» whib the other increases. We dsaU demolutratc thb
prc^nsition in a difiertot mantier afterwardr.

1O0. Cor. Wbeathesidesof thefeotaagie A£(;%;iei^ incrisas*
with uniform motions^ the rectangle flows with a motion that is
omfoAnly aecderalied : for therebta&gl^ is always e^ual to tha
sum of the triangle FDE and trapezium AFEL : bt>th of which
increase wttlrmotions that are uniformly accelerated, by cor. 4,
pvop.d; While the sides AD and AEracqoiietlKaii^gpMnuDG

VOL. L I and

ISO Of the Fluxions of Book L

gad LM, the rectangle A E is increased by the space DELMHG :
but there is no more than the parts EG and £M of this incre-
ment generated in consequence of the motion with which the
rectangle A£ flows at the term or moment when P. comes to
J), end Q to L. The part \b is generated in consequence of
the uniform acceleration of this moUon while the points iP and
Q describe D6 and LM. This part is rejected in measuring
the motion with which the rectangle A£ flows at that term ;
but the increase or acceleration of this motion generated in a
given time, or the uniform power by which this increase may
be conceived to be produced^ may be measured by it. A space
double of Ift would be generated in the same given time by an
uniform motion equal to the difierence of the motions -with
which the rectangles AH and AE flow ; and therefore the se-
cond fluxion of the rectangle AEmaybemeasured by thatspace;
as was demonstrated in the fifteenth theorem.

103. In the second case, the fluxion of the rectangle AE may
be expressed by the rectangle that is contained by LM and the
difierence of FD and AD, or of AF and 2AD; for that rec-
tangle is equal to the difierence of the rectangles EG and EM.
In this case, the fluxion of the rectangle AE contmually de-
creafies while AD increases ; it vanishes when AD is equal to
DF, or to one half of AF ; and the rectangle AE is greatest at
that term. When AD becomes greater than FD, the fluxion
of the rectangle AE becomes negative ; that is, the rectangle
decreases continually till it vanish, when AD becomes equal to
AF. The second fluxion of the rectangle is invariable, and is
measured by 2li, as in the first case : but it is considered as ne^
gative, because it continually diminisfaes the first fluxion, or re-
tards the motion with which the rectangle increases till AD be-
come equal to one half of AF, and accelerates the motion with
which the rectangle decreases after that term. If we suppose,
in the first case, the points P and Q (fig. 92) to move from O
and V towards A, and theu: motions to be continued aft«r P
comestoFandQ toA, we shall have the first and second cases
comprehended in one view.

104. 'We have insisted so much on the two preceding pro-
portions for the reasons mentioned in the 69th and 85th ar-
ticles.

Chap. 11. plane rectilineal Figures. * 131

trcles. In delivering those elements, we have endeavoured to
avoid every thingthat might appear abstruse or obscure, and to
obviate the objectiops that have been advanced lately against this
doctrine^ though we have not taken particular notice of them.
In the third lemma, we demonstrated that the motion with
which the triangle flows is perpetually accelerated while the
base flows uniformly. Nor do we see how this can be disputed^
since that motion cannot be supposed to be either uniform or
retarded for any partof the ttme^ how small soever. Some Philo-
sophers may be of opinion,,that a line or velocity cannot be con-
ceived to increase or decrease continually, but by certain small
indivisible increments or decrements only. How repugnant so*
^er this principle may be to what is demonstrated by Geome-
tricians,, the preceding propositions and a great part of this doc*
trine may be easily adapted to it, as we hinted in the 7th ar-
ticle. It was in effect in this form that some part of its elements
first appeared in the method of indivisibles : but we proceed to
establish it on the more accurate principles of the antients.

CHAP. III.

Of the Fluxions of plane curvilineal Figures,

LEMMA V.

105. J, HE base of any figure being supposed to increase urn-'

formly, the area flows with an accelerated or retarded motion,

according as the ordindtes increase or decreau while the base

.. increases.' . '- .

While the point P (fig.26) describes thebase by moving from
Atowards a,or from a towards A ; let the right hue PM, by mov-
ing pahallel to itself from AF towards af, cn^ from of towards
AF^ geoerate the 4rea APMF> or aPM/. Let ?D and PQ be
,. > I £ equal

^Tial parte of the base (fescrifce4fcyFi» ^
DE, mi QVf paraBd f& ¥Vt, Hieet tlte^ eurf*, e» toy e<mttet^ed
Itec FE/J m n, E, toi N : rtie» Jl 19 mcMilc»t^ tfe* whew the
ofdmates from jn to QN inereasej the ares 'DWHQ m greater
{ban the area DE/^J . iWefofe, when AP i^e^eoses tmifbnnlj^
or the point P describes jrDbefoife it ciese^ibes QD^acntftfi^right
ftfie PM generates tfie ai'ea 5 DE>i before it getterates 9QNE>
ffae ndotion with which the a»ea ffoW9 i» alwi^ aecelerated :
but if aP ittcrease cimformljrj of the point P more from Q to
q, the increments of the area aPMf which are geneirated tft vaf
eqnti times that succeed after one sAother^ perpetually cfeerease^
and the motion with which it flows is a^ays ietardied.

106. Itts evident^ in the same mannei:, lAaf, when t£te ordi^
nates decrease while the base AP d^reasesi the area APMF de-
creases with a retarded mo^n t bnt^ when the ordnra^tes in-
crease while the base aP decreases^ theareaaPMf decreaseswitb
an acceleieted motion. It is also manilSfest^ that^ wlien the area
APMF (fig. 27) is supposed toflow with an uniform motion^ s6
that any increments or decrements qnED, DENQ generated in
equal times are always equals then ^D is greater than DQ ; and the
base AP increases with a retarded motion^ or decreases with an
accelerated motion : but thebase aP increases with an accelerat-
ed motion^ or decreases with a retarded motion. It is easy to
represent the motion with which the area flows by the motion
of a given right fine pm, that^ by moriiig pwatt^ to ijtodf^ may
be supposed to generate a parallelogram always equal to the
area> as in the preceding chapter^ We omit this right line pm
in the following propositions ; but it may be easily sc^plied by
the reader^ i^ he pleases.

PROP, IV*.

107. T%eJiuxion of the base AD being rqfresetUed Ay^BO; the
fuxion of the area ADEF is accurate^ measured by thega'
ratteldgramlSlGk

E!i«t^lettlieliaseAJPO%.«§)^ incteamm3toim\f,ntkhn^^
iMLtBsPMiaef«isftwhiie4heba8ainci:^Qaa0i^ n»cradtoiMVi|tUui

case^

ClMp. in. phnu cumlimal Figures. ISS

case, with aq Accelerated raotimi^ by the last lemiDft ; abd^ if
this j;Qotion vas costumed luiifora^y from the te^rm or mGrnemX
wbiefx P coppes to B> a^aoe woidd be gencarated by it gt«9ler
than BDEC (bj ax. 4), bat lew thaa DGHE (by ax. 1), in
the MgoQJe time P describes DO widi its uiofonofi motion. Let
this space be s«{^osed equalto X^and XshaUbe eqaidto the
{isJFaflelograni £G. For, if it be said to be greater tha* £Q^
Jet DR begieater tbaa D£ in t^ same ratio; oHiiplete tbe pa-
rallelogf iMBEi DRLG> and it shall be equal to X, becaase it is to
EG as DR is to D£ (j^em. l, 6),or as X b to EG. Ut RL
ifus^ the cunre CH i& N^aad NQ parallel to D£ meet the base
in <j^. TbeOf since the parallelogram RG is to RQ as the ba^
X>0 is to PQ^ i(t f^QWB, from the first geaeral theorem (ait.
|6)^ that a ^aoe eqaaj to the parallelogram RQ wotfkl bege^
nerated by the motioaii with which the area APAIF fiowsj coo*
tiiwed unifpmUj f ixsop fiae tarm or moment when ^ coiftes to
J), in tba sai^e tin^ P describes SQ. Bat, whilie P deaciibes
^ line I>Q, the area DENQ is geaervted by the accelerated
motion with which the asea flowf \ aa^ D£NQmast be gx^al^
er ifajap t^eqrfoe which WQal4 have been generated in this titnc^
if the motion with which the area flows had been continued
ui|i£uinfy £ram the beginning of the time witiiouft anyaMde*
ration^by thaftrata^uom. Therefore the area D£NQ is ^eabST
thaa the payi^telogram RQ. B^t^ beeanse PR isgreater than
t>£ (in the sao^ proportion as X ^s supposed to be greater thaa
EG),and the ocdinates &am D|S to QJN continnally increase^
the parallelogram RQ is greater than PENQ. ^sid these be^
ing coatradictory, it follows tiiat the ifpaoe X is not ^eatef
than EG. If Xbesaid tp be less than EG^ let Dr be lesp than
P£ in the same proportian^ con^plete the parallelogram DrlQ^
and it shall jbe equal to X. Let r/p)nodaced meet d^e curve in n^
and nj paraM to PEmeet the base in jr. Then^ since the pa*
n^Uekgram rQ is to r f a^ the base PG is to Dq, it £)Uowsj
that a space equal to the parallelogFam rq would be generated
by Ihe motion with which the area flows^ continued uniformly
fron^ the tenn wh«i P comes to P^ in the same time P describes
a Unequal to P^ with its uiM&rm motion. But^ while Pde*
scpibe4 fP })efoie ^ came to P^ the s|^ace fn£P was generated

I3 tgr

134 ' Of the Flnxiohi of * 'Book 1.

by the accelerated motion with whidi the area flows ; and this
q>ace ^^ED must be less than the space which would be gene-
rated by the motion with which the area flows^ if it was conti-
nued nnifonnly from the term when P comes to D, by the se-
cond ^axiom ; that is, y«ED is less than the parallelogram rq •
which is absurd. Therefore the space X is neither greater nor
Jess than- the parallelcjgram EG, but precisely equal to it. When
the base decreases or increases uniformly, and the ordinates de-
crease at the same time, the area flows with a retarded motion ;
and the demonstration is the saine asthat of the 87 th article.
108. Let the area APMF(/?g. 27) flow uniformly, and acquire
theaugment DESKin any given time. LetDGbe the space that
would be described by P with its motion at D continued uni-
formly for the same time ; and the area DESK shall be equal
to the parallelogram EG. For, if the ordinates increase while
the base increases, the motion of P is retarded (by art. 106),
and if the area DESK besupposed equal to any paraUelogram
DRLG greater than EG, let RLmeet the curve EH in N, and
NQ parallel to ED meet the base in Q. Theti, because the pa-
rallelogram RG is to RQ as the base DG is to DQ, it follows,
that the point P, with its motion continued uniformly from D,
would describe the line DQ, in the same time that the area by
flowing uniformly acquires an augment equal to the parallelo-
gram RQ ; and the point P would describe a less line than DQ^
by the same uniform motion, whilethe area byflowing uniform-
ly acquires the augment DENQ. But because the motion of .
P is retarded while the area increases uniformly, it follows, from
Ihe third axiom, that the point P would describe a greater line
than DQ, by its motion continued nnifonnly from D, while
the area acquires the augment DENQ. And these being con-
tradictory, the area DESK is not greater th^h EG. In like
manner, it appears from the fourth axiom, that the area DESK
is not equal to any parallelogram Dr/G less than EG. There-
fore the area DESK is equal to the parallelogram EG. When
the ordinates decrease while the base either increases ordecreas-
fs, and the area is supposed to flow uniformly, the base flows
with 'an accelerated motion ; andthedemonstrationisthesame
as in the 89th article. In those cases, when the motion with

which

Ckkp, HL plane curvilineal Figures. 135

which the area flows is uniform, the parallelogram feG (which
is eqaal to the space that would be generated by that uniform
motion in a given time) is of an invariable magnitude ; and the
velocity of the point P is always reciprocally as the ordinate
PM : whence it may be determined, whether the motion with
which the area flows is accelerated or retarded, when the law
according to which the velocity of P varies and the nature of
the curve FEH are known.

• 109. In general, whatever the motion of the point P may
be, or that with which the base flows, the proposition is de«
monstrated universally from art. 107> by the ninth and eleventh
general theorems, in the manner that was described in the 90th
article. Therefore, when the fluxion of the base AD is re-
presented by DG, the fluxion of the area ADEF is represented
by the parallelogram EG. And, conversely, when the fluxion
of the area is represented by the parallelogram EG, the fluxr
ion of the base is represented by DG.

110. Cor. I. When the fluxion of the base AD is given, the
fluxion of the area ADEF is the same, whatever line be de-
scribed by the point M if it pass through E. If the base flow
tiniformly, whatever is generated more or less than the paral-
lelogram EG by the motion with which the area flows in the
time P describes DG, proceeds from the acceleration or retard-
ation of the motion with which the area flows at the term
when P comes to D.

111. Cor. II. When the fluxions of the bases of two figures
that are generated in the same time are equal, the fluxions of
the areas are as their ordinates ; and when the ordinates are
always in a given ratio (as in the case described in the intro-
duction, page 1, fig. 1), the areas that are generated in the
same time are in the same given ratio, by the sixth general
theorenv. When the fluxions of the bases are reciprocaUy as
the ordinates, the fluxions of the areas are equal ; and when
the fluxions of the bases are always to each other in this ratio^
the areas that are generated in the same time are equals by
the fourth general theorem.

112. Cor. III. Let AE (fig. 28) be a rectangle containedby the
two aides AD, AL; while AD and ALflow with any accelerated

I4 or

1

134 ' OfikeFluxiMsof ' -Book L

by the accelerated motion with which the area flbws ; and this
qMtce qnED must be less than the space which would be gene-
rated by the motion with which the area flows^ if it was conti-
nued nniformly from the term when P comes to D, by the se-
cond ^axiom; that is^ qnBD is less than the parallelogram rq:
which is absurd. Therefore the space X is neither greater nor
.less than> the parallelogram EG, but precisely equ^ to it. When
the base decreases or increases uniformly, and the ordinates de-
crease at the same time, the area flows with a retardcfd motion ;
and the demonstration is the same as that of the 87th article.
108. Let thearea APMFOfg. 27) flow uniformly, and acquire
the augment DESK in any given time. Let DGbe the space that
would be described by P with its motion at D cbntinued uni-
formly for the same time ; and the area DESK shall be equal
to the parallelogram EG. For, if the ordinates increase while
the base increases, the motion of P is retarded (by art. 106),
and if the area DESK be supposed equal to any parallelogram
DRLG greater than EG, let RLmeet the curve EH in N, and
NQ parallel to ED meet the base in Q. Theri, because the pa-
rallelogram RG is to RQ as the base DG is to DQ, it folloH
that the point P, with its motion continued uniformly from D,
would describe the line DQ, in the same time that the area by
flowing uniformly acquires an augment equal to the parallelo-
gram RQ ; and the point P would tlescribe a less line than DQj
by the same uniform motion, whilethe area by flowing uniform-
ly acquires the augment DENQ. But because the motion of
P is retarded while the area increases uniformly, it follows, from
the third axiom, that the point P would describe a greater line
than DQ, by its motion continued nniformly from D, while
the area acquires the augment DENQ. And these being con-
tradictory, the area DESK is not greater th^ti EG. In like
manner, it appears from the fourth axiom, that the area DESK
is not equal to any parallelogram Dr/G less than EG. There-
fore the area DESK is equal to the parallelogram EG. When
thcordinates decrease while the base either increases ordecreas-
fs, and the area is supposed to flow uniformly, the base flows
with 'an accelerated motion ; and the demonstration is the same
as in the 89th article. In those cases, when the motion with

which

Cha{». TIL plane curvilineal Figures. 195

which the area flows is uniform^ the parallelogram EG (which
is equal to the space that would be generated by that uniform
motion in a given time) is of an invariable magnitude ; and the
velocity of the point P is always reciprocally as the ordinate
PM : whence it may be determined, whether the motion with
which the area flows is accelerated or retarded, when the law
according to which the velocity of P varies and the nature of
the curve F^H are known.

• 109. In general, whatever the motion of the point P may
be^ or that with which the base flows, the proposition is de«
monstrated universally from art. 107^ by the ninth and eleventh
general theorems, in the manner that was described in the 90th
article. Therefore, when the fluxion of the base AD is re-
presented by D6, the fluxion of the area ADEF is represented
by the parallelogram EG. And, conversely, when the fluxion
of the area is represented by the parallelogram EG, the fluxr
ion of the base is represented by DG.

110. -Cor. I. When the fluxion of the base AD is given, the
Huxion of the area ADEPis the same, whatever line be de-
scribed by the point M if it pass through E. If the base flow
uniformly, whatever is generated more or less than the paral-
lelogram EG by the motion with which the area flows in the
time P describes t)G, proceeds from the acceleration or retard-
ation of the motion with which the area flows at the term
when P comes to D.

111. Cor. IL When the fluxions of the bases of two figures
that are generated in the same time are equal, the fluxions of
the areas are as their ordinates ; and when the ordinates are
always in a given ratio (as in the case described in the intro*
duction, page 7^ fig. 1)> the areas that are generated in the
same time are in the same given ratio, by the sixth general
theorem. When the fluxions of the bases are reciprocally as
the ordinates, the fluxions of the areas are equal ; and when
the fluxions of the bases are always to each other in this ratio^
the areas that are generated in the same time are equal, by
the fourth general theorem.

112. Cor, III. Let AE (fig. 9S) be a rectangle containedby the
two sides AD, AL; while AD and ALflow with any accelerated

l4 or

}89 qfihefUxiofii^ ^ Votkf.

pr xf^fixifA no^iKmSj H B ifi¥:x\h» wj Ijiive UK ^ in^toMe
^ the sk^^ 4-D in ?• T^en^ tb^ rectiingle A£ l^ing eqmJ to
Ijbe ^um or diffearence oftb^ ^eiis AFEL md f ]>£> th^ftiqQmi
p{ the jreojtimgle AE dk^ )^ mea^msd )>y ibe siw or dilfeimee
of jijbie ^ifiict^gle^ ]EG mi £Mi whm the A^xjoQs^f the si46»
AI^^ A{ij arp<aem^r^ hy ihe right lio^s IHGr aad LM. Azu) dw^
tj^e ij^fl c^iae of ihe tbkd propositiim i» briefly devioiiftettted.

1 13. Cor. TV. If the area AD^^QSg. 39) be alweyfttcipwltpthe
30ftol|ki^^ .€)0p^ltfPf^ by the ordinate D& fun^ the right line J>6
tWtijepffysnis tbe flmdon of the bene AjD> then ^^udl jthe p^t
r&Ue]pgr£m icQ ^^QLe^i^sure jdie fliqdon of tiie ere» ADrf, or <^
the p«ar0lWlogra9i £G, or the se^md fluxioa of the sma A2>£F,
Aii^dj if t^he ^irea AiDef be always equal t^ the fMreUelognwa
{Co the parallelQgrain eG «bidl measure the tiiird fluxiou of
the em» AD£E In the »we manner tb^ bigfeer fluxions of
diiifi iap» ipay be ]:e{»]«sMte<ji ; but^ inio^ieoiiwBj by proceed^
ing thus^ we shall come to sQme fliudofi tbfi^ is/cmaiafit^ l^ueh
«^Bi)ot be coD^ideii^ «s » flowing or ym«iMe quMliiy.

) 14* CprV. Iix»»^^tbi9 pfoppBition 9pnie^nter«l theoienw is
fib^^ocbrineofmotipn may )^ easily demo98irated* IfthebaM
4P 0l^g-d6 jcao) rq>rese9t the time ofeny motimiuPMtheTelor
city M wy t^rm of the time P> the^acedeaoribedby theinotidii
in the ti«^ APmey he qaueasoxed by the area APMF; thatia^the
ipa^ dnsciibedift the time AP shall he to die «pa« described
in any other time AD as the area APMF ]$ to the uroa AO£F.
fqfj tbe motion of P benig unifoimj the motion with which
ttie »ea APM^ fl^vs yiuries in the same proportioii m Ihe or-
dinate PM> by vhMhis been jdemon^trated ; and therefore, if
the invairtable right/ line pm, by moving j^mM to its^, de«
imk^ a f pitce ant Always equal to the asea APMF^ the velociiy
of the n^t Une ;?»»; o^ of the point p, shall be always meaMsr^
D^hythe Qi4iPiifaePM at any term of the tiiae, as P. From
whi^ a foUras, Qpav^erteLy (as m act. 9£)> that if we suppose
J^e y^caty of the fsmip at any term P to be measured hy
the orclinate PH, the apace desaihed by the point|^ in the
time AP shall be measured by the area APMF; Of, the spaoe
4escrihed byjpin libe tmeAP shall be to the space descrihttlby
itmthetttneADastheareaAPMFiatotheareaADEF. Lt

the

T\iteK/^/X>.VffU.

I — 7"

A

li /^.u.

^— T '

DO TO

4

. I

Chap* III* plane iurviUneal Figure9. t9t

the aane maimo; if 4lie aotion or exerticn o£ aay fawBt Aitt,
ModtiorUMB or lelaids jnotiun be alvays BefMresentedtiy the er*
dinate PM while the base AP represents the time^ the motion
that is generated or destroyed by this power in the time AP ia
to the motion that is generated or destroyed in the time AD as
the area APMF is to the area AD£F; Quantities of all kinds
t^at increase or decrease while the time increases in any xegulnf
manner whatsoever^ may be thus sulyected to mensuration by
the qua^jbr^ture of figures.

1 15. Cor. VI. Here is anoAxer general theorqn in the doc-
trine of motion that also follows easily from this proposition. Let
the base AP (fig. 27) represent the space described by any mo-
tion ; let the ordinate PM be always reciprocally as the velocity ;
thai isylet the ¥c4oeity at any placeP be always tothe veloeity at
aoyolherplaceDastheoFdinaiteDEifitotheoidtRatePMs and
the tune in which the fine APis deisoibed shall be to the time in
which AD is deseribed as the area APMF is to the area ADEF«
Fox it follows fiom this proposition^ (art. 108) that^if the atea
APMl^ flow uniformly^ iAxe velocity of the point that describes
the base shaU be reciprocally as PM ; that is^ its velocity at P
shall be ta its velocity at any given term D as DE is to PM.
From which it easily fellows^ conversely (as in art. Qi), ^at,
if the velocity of a point tlmt describes the line AP at any term
of that line^ as P^ be idways reciprocally^ as tiie ordinate PM^
die area APMF shaH flow unifonnly^ or in the same propor-
lioH as the time ci the motion.

1 16. Let a given right Une SP^ revolving about the cen-
ter 8^ g^erate the circle ADa, and meet any curve FEH al«
ways in M ; then, if the ray SM increases while the ardi AP
hiGPeases, it is manifest, that, when the motion of P in the cir^
osmference ADa is uniform, the area FSM flows with an acce-
lerated motion. For, if BD and DG be equal arches described
by P in any equal times, and the rays SB, 8D, SO meet the
curve FMH in C, E, and H, the area SEH shall be greater than
SEC. The sector ASP and the angle ASP increase uniformly
hi this case. If the motion of P be variable, but its velocity at
Bhe measored by the ardi DG, the motion with which the sec-
tor ASD flows shall be measured by the sector DSG^ and the

motion

158 Of the Fluxiom of \ BoakL

noliffli with which the angle ASD flows by the angle DS6 ; as
maj be <kmoDstrated in the same manner as the 60th article.

PROP. V.

117. The- area FSE (fig. 31) being supposed to flow as in the
last article^ let the arch EI described from the center S meet
the rm/ SG in I ; and^ if the fluxion of the sector ASD be re*
presented by tlie sector DSG, the fluxion of the curvilineal
area FSE shall be accurately measured by the sector ESh

Let the motion of P in the circumference ADa first be uni-
form, and the rays SM terminated by the curve FMH increaae
while AP increases, as in the last article ; and the area FSM
shall flow with an accelerated motion. Suppose that a space
equal to X would be generated by this motion continued uni^r
finrmly from the term when P comes to D, in the same time P
describcsDG ; and the space X shall beless than the a^rea £SHy
(by ax« 1), but greater than the area CSE (by ax. ^. If X
be not equal to the sector SEI> let it first be greater than.lhat
sector. Produce SE to R till SR be to SE in tlie subdupUcste
ratio of the. space X to SEI ; let the arch RjL described froia
the center S meet the rays SD, SG in R and L^ and the curve
in N ) and the space X shall be equal to the sector SRL< Let
SN meet the circle in Q ; and since the sector SRL is to SRN
as DG is to DQ^ it follows (by theor. 1% that, if the motion
with which the area FSE flows be continued unifcurmly^ it will
generate a space equal to the sector SRN^ in the time that P
describes the arch DQ. But tliis space must be less thi^ the
area SEN which is generated in the same time by the accelerated
motion with which the area FSM flows^ by the first addom.
Therefore the sector SRN is less than the area SEN. But this
is absurd ; and, consequently, the space X is not greater tha^ the
sector SEI. If it be said that the space X is less than the sector
SEIJetSr be less thanSE in the subduplicate ratio of the space
X to the sector SEI, From S as center describe the arch rl,

meet«

Chap. III. plane curtilineal Figures. IS§

meeting the rays SD^ SG in r and I, and the ciurve EC in n.
Let S» meet the circumference of the circle ADa in q ; and the
space X shall he equal to the sectgr Sri. But Sri is to Sm aft
rl is to m, or DG to Dq ; and therefore (by theor. l)j a space
equal to the sector Sm would be generated by the motion With
which the area FSE flows continued uniformly^ during the
time in which P describes an arch equal to qD ; and this space
must be greater than the areaSitE which is generated by the «iq-
celerated motion with which the area flows while P describes
qjy before it comes to D, by the second axiom ; that is^ the sec*
tor Snr must be greater than the area S12E : which is absurd.
Therefore the space X is neither greater nor less than thesec«
tor SEI^ but is precisely equal to it.

1 18. The rest remaining as in the last article^ suppose the area
FSM (fig, 32) to increase imifonnly^ and to acquire the equal in*
crements CSE^ESH in any equal times that immedi ately succeed
after each other. Let the rays SC and SH meet the circumfer-
tnce ADa in k and K ; and, since the rays from S terminated
by the cun^e CEH aresupposed to increase continually from SC
to &H, it is manifest^ that the angle ESHis less t];ian£SC^ and
that the arch BK is less than Dk. Therefore the motion of the
poiniP,in describing the circumference A'Ddj is perpetually re-
tarded. Suppose that the point P would del^cribe the atch DG
by its motion at D continued uniformly^ in the same time that
the area FSE by flowing uniformly acquires the augment ESH ;
and the area ESH shall be equal to the sector ESL For, if
ESH be greater than ESI^let SR be greater than SEin the siib-
duplicate ratio of ESH to ESI ; let the arch RL described from
the center S meet SG in L, and the curve EH in N ; and let
SN meet the circumference DG in Q. Then, since the sector
SRL is to the sector SRN as the arch RL is to the arch RN>
or as DG is to DQ, it follows, that, in the time the area FSE
by flowing uniformly acquires an augment equal to the sector
SRN, the point P would describe DQ with its motion conti-
nued uniformly from the term when it comes to D ; and, in the
time that the area FSE acquires the augment ESN (which is
less than the sector SRN), the point P would describe an arch
less than DQ by the same motion continued uniformly. But^

while

■w-n

)4p Of the F2wiom <f BookL

while ^ area FSE acquires jthe augment £SN^ dbe poii^ P
witb a retdrjde4 lDotJk}^ d^stcribes DQ ; and it would ^e^be
ap aFcbgieal^ thitn DQ ip the asme time b^r its motioii contip
][iued uniibnnty frojn j), by the third axiom. And these heiog
qontraij^tp^, it foUows, that the area ESH is not greater thaa
the sector JBSI. Ju the saote xnaniiier it lappears^ that J^H ji^
ppt equal to any sector Sri less thw SEL

119. In genera]^ ajl the oih^r cases of this proposition ^e
deduced from the first, by the i^nth and eleventh theorenu^ ia
the samemanner that we demonstrated the 90th article : so iiml
when the fluxion, of the arch AJ) is represented by DG, or the
lbi;cioA of the sector ASD by the sector DSG^ the fluxion ^ tbe
area FSE is expressed-accuratei^ by the sector BSl ^ and^ cqat
Tersely^ when the flujidaa of the area FSE is expressed by ihe
seetor BSI; produce SI till it meet tlie circumierenoe ^Do P^
Q, and PG shall r^eaent the fluxion of the arch AD.

120. Cor. I The fluxion <»f the sector ASD is to the Buxii^^
^f the are^ FSE in the dt^hoate ratio of SD to S£, When th^
arch AD, or the apgle ASD^ flows, uniibrmly^ the flu:i^if>D of
^e ar«a FS]E1 increases or d^ecreoses in the duplicate ratio ef ib§
^y SiEL But^ when the a«ea FSE flows unifondy^ the fluxioQ
of the arch AD^ or of the angle ASD^ is jreciprocally «a tbi^
s^Qore of the ray S£.

liJ. Cor.ll. (JigS^)f In Uie 1 16th article>.wherethe area waf
apposed to flow uniformly^ it was shown^ thal^ if the sector ESi
he eqttal to th< area ESH^ and Slp^pdueed to thecircumfeceQce
ADa meet it in G, then DG is the arch that wcmld be descrSi*
ed by P with its motion at D contiiiued uniformly while the
area FSE acqairea the augmeiit El^* From thi^ it lollowsi
that if the angular nation of the ray SE was co9kinQe4 ^i<»
farmly^ th^ angle ESI would be generated by it in the same
time. Let SH meet the arch £1 in R ; and the ang}e ESB^
which is actually generated by the ray SP revolving about the
center S while the area ia increased by ESH^ is to the angle
ESi that would have been generated in the saojie timtif if tb^ an-
gular motion of S£ about S had been continued unilbrmly^ aa
the sector ESR is to th^ aroa ESH; and the difierence of those
ao^es is to the latter angle ESI w ibe ana ERH is to ^e aiea
t;SH. l««. The

Clkap« til. plum curvUkual Ftgf^rnt 141

12ft. The amikvify of cnsvilincal ftgOMis may be dnriio^
ftom duii^ q£ icotalmeal figuics that 4re ahrajs sittilarly describe
edin ihem; or^wenaycompiehendaHaOTU of fiflHiiirfigeMi
pliemes, or solidsj in this general definition. Figures are similar
when they may be supposed to be placed in such a manner,
that^ any right line being drawn from a determined point to the
terms that bound them^ the parts of Uie right line intercepted
betwixt that point and those terms are always in one constant
ratio toeteh otker. Thns the figures ASD'O^. d5)> aS(2are simi-
lar^ when, any line SP being drawn always from the same point
S meeting AD in P, and ad in p, the ratio of SP to Sp is inva-
riable. It is manifest, that the rectiU jle^l inscribed figures APDS,
apdS are similar in this case, according to the definition of such
figures that is given in tbe Elements^ If the right lines FM
and jm. pajraHel to eacb other meet SAt in M and m ; theft PM
sha}l be to jpm as SM is to Sm, or as 9A i& to Sn ; and AM
is to iKne as PM is to pm* And, conversely^ if AM be dways
to am as AS is to aS ; and PM be always to pm as AM is to
am, it is manifest, that the points ^,p, and P are always in one
rigfiftBne, arid that Sp is to SP in the invariable ratio of S« to
SA. From which it appears, that the common definition of
similar figures, paflicularly that of jfpollonius for the conic
sections, and of Cavalerius for any figures whatsoever, is in*
eluded in this, or is easily deduced from it. When the simi*
lar figures are in the situation we have described, they are also
similarly situated, and all their homologous lines are either
placed upon one another, or parallel.

1^. If flie right line SP, by revolving about the given point
S, geneMe the similar ffgures ASP, aSp, their fluxions shall
he to each other iA the duplicate ^atio of SP tol^, or of SAto
Sa ; and, consequently, in an invariable ratio. Therefore, by
the sixth'general theorem^ the similar figures ASP, aSp which
are generated in the same time are in the same invariable ratio.
Theiriaiigle& ASP, a^ and the segiftents ANP, anp are in the
sailtte derplicate ratio of SA to Sff, or c^ AF to ap. If we sup-
]^ose tht ray Sa and the figure aSd to inx:rease or decrease so
as fo remain always similat to a given figure ASD; the fluxioii
of tlie area ttSi shall be to the fluxion of the inscribed triangle

aSd

]4£ Ofihc fluxions of SoHds^ . Book L

«S«i in the invariable ratio of that area to the inscribed tri-
angle ; and the jfluxion of the area aScfmcreases or decreased
imiformly when the fluxion of Sa is constant*

CHAP. IV.

»

Of the Fluxions of Solids, and of third Fluxions.

PROP. VI. '

124. Jl HEJluxion of the solid that is generated by the revo^
lution of the area ADEF about the axis AD (fig. ^6 J is
measured by the cylinder that is generated by the rectangle
E6^ when thtfluxionofthe axis AD is measured by DG.

The demonstration is the same as that of the fourth pro-
position ; and it is easy to deduce corollaries from this propo*
sition similar to those which were inferred from the fourth*

PROP. VIL

125. The fluxion of a solid that can be conceived to be gent*
rated by any plane surface moving parallel to itself, andper^
pendicular to a given axis, is measured by a prism that has
the generating surface for its base, and its altitude equal to
the right line which measures the fluxion of the axis.

By a prism weunderstand^in this proposition^ any upright solid
tliat can be generated by an invariable plane moving parallel to
itself perpendicular to a given right Une. When the generating
figure isof an invariablemagnitude^thepropositionisdemonstrat-
ed as in art. 80. The axis AD being supposed to flow uniformly,

if

Chap. TV. Of the Nuxions of Solids. 143

if the generating plwe LM (fig. 34) continually increasej it i3
evident that the solid must increase with an accelerated mo^
lion. Let this motion be supposed to be continued uniformly
£rom the term when the plane LM cuts tiieaxis in D^and toge*
nerate a soUd equal to R in the same time that the axis by flow-
ing uniformly acquires the augment DG. Then shall thejsolid
R be equal to the prism upon the base LM of the height DG:
for, if the solid R be said to be greater than that prism, let AD
be produced to P till Im (the section of the solid through P
parallel to LM) be to LM as the soUd R is to that prism ; and
the solid R shall be equal to a prism upon the base Im of the
height DG. From this it follows (by tbeor. 1), that the mo-
tion with which the solid EBML flows, continued unifoirmly,
would generate a solid equal to a prism upon the base Im of the
height DP, in the time that the axis acquires the augment
DP by flowing uniformly. But the motion with which the solid
flows when continually accelerated fromi the same term, gene*
rates in that time the firustum included betwixt the sections LM
and Im, which is less than the prism upon the base Im of the
height DP; and, if the motion with which the solid flows was
continued uniformly from that term, it would generate in the
same time a less solid than that fiiistum, (by ax. 1), and there*
fore less than the prism upon the base Im of the height DP.
But these are contradictory ; and therefore the solid R is not
greater than a prism upon the bas6 LM of the height DG. In
like manner it is shown, that the solid R is not less than this
prism. And the demonstration is extended to the other cases
ofthe proposition in the manner that was described in art. 90.
Therefore, the fluxion of the axis AD being represented by DG^
the fluxion of the solid is accurately measured by a prism upon
the base LM of the height DG. And, conversely, when the
fluxion of the solid is represented by that prism, the fluxion of
the axis is accurately measured by DG.

126. Cor. 1, Let the solid ALM (Jig. 3o)beapyramidthathas
its vertex in A, and let the section LM be triple of the square of
AD: thenshall thes(did ALM be always equal to a cube describ-
ed upon AD. Thi^efore the fluxion of the cube upon AD is
accurately measured by a pai-allelopipedon upon a base that is

\ triple

144 OftkeFlmwmefSeflulk Bookt

triple of the sqaare of AD, of a height eqnal to D6 that wut^
mresi the Andoa of tlife side AD, ot by a paraUddpipmloii vpiM
l&e scpiaie of AD of vh^ighit) trij^ of D6. From i^Aiidt it
foHd^s^ tkiKl, wIko foar quantitiev are in eontinaed propoitH>B>
and^thci fiint tam is invariable, the flanon of die secoad tena
ia tao&d Aird part of th^ flaxionof tiie foaffth as the first term
is to die diird<. %

It?. Cm*. IL Wheik the solid ALM is a pyramid as in tfatf
Usfi tmoHhijr^^d the motion with! which AD flows is onifcnii,
diaflnnowof the pyradiid is as theseadom* LM, or oes the sqnare
oft AD. I» this case, the right lines AL, AM, Sfc. that form
the solid im^e at A, flow with> ahiiform motions. The sjsction
hSM, and the triangular sides of the pyramid, increase with
motions that are aocelersAed uniibrmly, by prop. 2, eor. 4. But
the solid ALM' increases vnth a motion the acceIa:iation of
Vfhich increases nnifocmly ; that is^ the second fluxion of the
pyramid incmB»es mufbrmly, and its thiid fluxion is invariable.
In the same manner, ^vtien the skle of a cube or the axis of a
col^e incfeartes^ miifbnttr}y> the solid flows with a motion the
acceleration of which increases unafermly.

128. In art 93, we resoked the trapezinm DGH£ {fig. 21)>
(which is the increment of the triangle ADE that is genemted
iHifile AD by increasing n^formly acquires the augment B6)
into two paits, viz. the parallelogram EG which is generated
in consequents <^ the motion with which the trian^ flows at
the term when P comes to D, and the triangle EIH, whidi ii
generated in consequence of the aoceleratioQ of that motion*
In like manner, we are in the present case to resolve the in-^
crcment of the solid that is generated while the axis acquires
the ar^gment D6 (Jig. 35) into three parts. The first we con^
eeive to be generated in consequence of the motion with
which the solid flows, at the term when its side or axis becomes
equal to AD, supposed to be continued uniforaily for that
time. The second part is conceived to be generated in con-
sequence of the acceleration of this motion supposed to be
continued uniformly from the same term and for the same
lime. And the third part is what is generated in conseq^nce
of the continual ainl uniform increase of this aoceleration.

129. These

Chap. tV. q^tkiti tbisitmi. I4«

190. These tkneparUnuq^ be distiiigiiisbedfiome^
in the foikwing maimer. Letthelria]ig}eMBfli(^g.36)>bymor-
ing pci)pei|4iciilar to the li^t line AP^ geqerale the p jnramul
AJrla HI. w nue tnepomt ir oeacriaes JAxjiet tne pyramMi aoquie
themerfmentthaikteradnatedby thepjanes EDe, HGAperalldi
to UPm; lei BI and a parallel to D6 meet OH ahd gk in I
and 1^ and eN parallel to EH meet HA in N. Then the frus-
tom of the pyramid tenninated by the planes EDc^ HG&»re-
Bolved into tbjcee parte ; the prkm WDelQi, the f^^im EIHet N^
and the pyramideNtA. The fint of these^ the prim EDelGi^
measwes the fiist flaxion of the pyramid ABSe when 1X> re-
presents Ae fluxion of AD^ by the serenth popesition. Hie
seoond^ viz. the prism ElHdif, measaies one half of the flax-
ion of the fffst^ the aUitade DG being supposed invarieble*
For the fluxion of the triangle tiDi, or lOi, is »easared by the
parallelogram IN (the fluxion of DE or GI being measmed by
IH}^ by prop. %, and the fluxion of the prism ED^IGt is there-
fore measaxedby » paraUdapipedonNHEf «pon the base IN of
the ^tode DQ, which is double of the prism EIHetN that is
t^>on die same base IN and of the same altitude. Therefone
the prism EIHeiN measures pne half of the ftixionof the pntm,
EDf IGi, or onehalf of thesecondfluxionof the pyramid ADEe .
The last part of the firasSum ED^HGA, viz. the pyramid e^ih,
measures one sixth part of the fluxion of the parallelopipedon
KH£«. For^ ccHnpletin^ the parallelogram NiRA and the pa-
rallelopipedon NALe^the fluxion of th^parafldograin IN is
measured hf the parafielogram NR (by prop. \, IH being in«
vurialile)^ and the fluxion of the paraMopipedon NHEe is mea-
sured by thepacaHelc^ipedon NALr^of whidh the pyramid eNih
is one staetb part. T}ierefore this pyramid measures one sixth
part of the fluxion of the pardQelopipedon NH£e> or of the
third fluxion of the pyramid ADEe.

ISO. nmstbe solids £D<?IGi,NHEf, NALere^^tivelymea'
sure the ^mt, secondhand tiwrdiluxions of tbepjmmid ADEf^
whenADflows aflifermly> audits fluxion islepi^ented by DG»
The &ree parts iMhich omstitatethe frustum EDcfi^A^or^dste
incn-ement <^4ihe {^amid that lagaaerated while AD acquires
theaugm<snft D&Xare^ the prism £I)elGi',4h^ntiftm EHIeN^j
aad^liepyramideKii; Qii^ch the first mesOnomitheflmfluzfi

VOL.1 K nm.

146 ,0/ third Buxiata. . "Book!

ion^ the second measures one half of the seboxkl fluxion, the
last measures one sixth part of the third fluxion of the pyramid
ADEe; and this part eNiAbeipg invariable^ the pyramid has
no fourth fluxion. The first of these is the solid tliat would
have been generated bj^ the motion with which the pyramid
ADEf flows^ if it had been continued uniformly for' the time in
which P describes DG. The second part is whatiwould have
been produced more than the first, if the acceleration of the
generating motion had been continued uniformly from the term
when P ccmies to D, in the same manner as the triangle EIH
(fig. 21) is the space that is generated in consequence of the
acceleration of the motion with which the triangle APM flows,
while P describes DG, according to what was shown in the 93d
article. The third part e NiA is what is generatefl in consequence
of the continual uniform increase of the acceleration of the ge^
Derating motion, or of the uniform increase of the power by
which we may conceive this acceleration to be produced. Tlie
first fluxion of the pyramid ADEe is accurately expressed by the
first of those parts of its increment, and the other two are just-
ly neglected in measuring it. The second part serves for com-
paring the second fluxion of tlie pyramid at the term When P
comes to D with its second fluxion at any other term, or with
the second fluxion of any other solid. The third part eN/A
serves for comparing the third fluxion of the pyramid with that
of any other solid, or witli the third fluxion of the same pyra-
mid when the axis AD flows with a different motion.

131. The prism EDcIGi, which measures the first fluxion of
the pyramid ADEe, is to theparallelopipedon NHE«, which
measures its second fluxiQn,««s one half of DE is to £K, or as
one half of AD is to DG. The parallelopipedon NHEe is to
the parallelopipedon UhLe, which measures the third fluxion of
the pyramid ADEe, as NH i? to NA, or as AD is to DG. And
It is manifest, that tliese ratios, by dimipishing D£, or.bydimi-
iiishing the increment of tlie pyramid. EDeHGA, may become
greater than any assignable icatip. Therefore the ratio of the
quantity that measures the first fluxion pf the pyramid ADEe
pq that which measures its;(econd fluxion, and the ratio of the
]j^ei to that which measures, it^ third fluxion; may become

... greater

. » A

n

greater than my ajfeigiiable ratUbj when the ijoiuitity that mea-»
sures its first jBuxipn is oontinually diminished. The pyramid
A6HA is pxeciaely equal to the sum of these four solids^ the py-;
ramid ADlS^, the prism EDelGi, the prism EHIrNi^ and .the
pyramid eHih. When D6 is continually dimtxiisbed^ the first
x>f those four solids approaohei to the pyramid A6HA, so that
theirdifferencemaybecpmel^ss than any assignable magnitude;
hut the smn of the first two approaches much nearer to the py-
ramid AGHA ; and the ratio of the difference betwixt this sum
and that pyramid^ to the difference of the pyramids AGH A and
ADEc,mBy become less than any assignable ratio. The sum
.of the first three of those solids approaches still nearer to the
Y^ue of the pyramid AG HA ; and the ratio of the difference her
twixt that sum and this p3rramid/to the difference of the sum of
the first two sdi^s and the same pyramid^ may b€fcome less than
any assignable ratio. Thus we are led to approximations of va^
rious d^rees of exactness by this method^ as well as to accu-
rate mensuratiofis. For^ in general^ the value of a fluent at the
beginning of any small time being given^ we approximate to
i^ value at the end of that time^ by adding to the former^ firsts
the quantity that would have been produced in this time by
the generating motion^ if it had been continued uniformly from
the beginning of the time (which we suppose to measure the
first fluxion of the fluent); then one half of the quantity that
measures its second fluxion^ and one sixth part of the quantity
that measures its third fluxion. The approximation is the more
accurate, the more we add of those quantities together, in the
order we have described them : butof this we shall treat more*
fully afterwards.

.132. When AB increases uniformly, the prism KDcIGi in-
creases in -the same proportion as the triangle EDe.(it|i| altitude .
IXjr bei^g mvariable), or in the duplicate ratio of AD; rjtfae pa-
rallelopipedon NHEe increases in the s^^i^e proportion as £<s or
AI>increa8e8^ because the Bi<)es HK and HI are invariable; the
parallelopipeciQiB NALe is always of the 3ame magnitude . But
when AD ifr giyen^ and DG, which measures the fluxion of
AD, increases ox decreases, the prism EDelGi, which measures '
the first fluxion of the pyramid ADE^ ^ increases or dp^rease^ in

K2 the

Ui Ofthif^ ttuitioksi lMk\.

[tilled llife second ilM^ of tlyiA;fr5%4|^

ChiMMxtM of lihe sdAe ffimAis i^tit«ines/^r d«M^^^ A Vbk
trij^tete Vatid of ei, <fr D6. TMteftH^ ^;vMlft Xftt ^km lkii&
ion bfttife^yraiiitd «t any gi^M t(Ml4& MfEp(Mft V6 ^m^> %fa«
secoftA fluxion Varied fti the AipttcAife^kMlii^ ttM 'llk^ VAM ttn^
mih the MpOcaC^ l^tio of thilttewU^hlli^ftist A*tdMi «M^..
1 33 . AH we haVe^id'€t1(he)^yk'M^i/i At>Sfc <^/.«^.li;HMa%
fiukiitesis e«»ily transfeltedto lAe edti^/iiAMEbiAMi^dM.hy l^ltttt
Wte^aemonst¥ated ih the ihtfdd6t«akfti'(|>^ t9 <t ^6). Thb itt^
t^'eAent df the cone AHA OifM is^lkerfiit^ ¥4^8^ IM^ ttd^ ^E
IS focrea^edl)^ EB, or the kb^m mcrWeSi «y ^A% «[^^Nite
CBGH revolvfog 'about «he ^kis £B, is ^equtil (^ iSMe t^yRnO^
YXxy (BX "being a nlem pfr o)>d/ei6niil be««i^M: 'Ite "itad EH)
-and the doneES/(BI^ being the^iffbrence^fiC^MC^ taken
^^^tli^r. l^ife cyitnaer Y^ Kih^ ¥eilolY«a«)R» «^pU«i,
<the cJrlihJter HZrA, NiiaWfeliollo«r cylindric -solid ftat is:ge-
H^t^a'by the rectangle itX^ervolVfeg fcWift^hte a^EB ;
^e4irst df these Ttiea^irts the ftiisft lloxlon «PflA fceita^ AHA,
Hv-hAi theaxrs fiotv^ tmtfdrtnfy, tfna its firit =ftitoci«fife¥^i'esdrit-
edbyEB; dndthelatter'pttrtii^e^sHite9ohe^alf^«lhe1hi:ri6h
of ^hefornier, or dne half ^fShe second fWxidh WWe c6hfe
iA^h. For, <heflhxion df^EH'beHtg-tepiWdfted hyXTZ/i^^
Iffieflnxion of its sqiiare l>y %he>t«(%dh^le tfd^UdUeB % ^H
and CZ '(W'|wr6p. ^, c6r. ^, ^he ^iiffir^neeiif the «tt!aJ* of
^X and BZ (dr^e'tfecttiiigfe cSdWiiiM'bj'^'aiia OE)-!ilca-i
snres one half of the fluxion of tlie square of Bft i 'iSnS'^&fit ah-
tUiWipade desc5Hbcd by ZX i^bhHfi^ tlbAtit B^^^sa^'one
halfbfflie'flti'xiohofthe^iH^fed^driKa^'B^ Th«l«-

fift-e dle-h6lk^ Mtid g^f^t^t^dby fiiMreaangfc^XWebMyes
M:^ 'h^ of ^he^fltt^^ibn df Ihe f^fibdfo 4Ilfe2/ttr 'Mie Kalf ^f
the^condHttxittii of *he dwie AH*. Tlife <Joi^«yC*'W<Si,
'^ith the t^^Httdbr YXi^, ctfhiplfelibs'a scMM i^M'fb^tMe^f^^
*H«cG) m^sures otte'thitd p6f 1 6f^lhe«d3ribh'«f *ie^<ftft«r st>-
iid'a^t^ibea-by therect^eWEX ; bfet^iiite Ote'BtHtfdh'oiEHhe
f^«fsrdf'tfie 'sq>J^re -ofBK -alHy^^4hfe-»qu*l:t W qte-(6r-<^

fluxion

I

dhftfuIV. QftiMMmwL 149

4«a»i«ii 0f liti^iMitMi^cmtiaiiecl Ij^ £H %TidCZ) is vneasiuBd
by the square of BZ Of of C$( ftpd ibf flo^vm of tbe aa)id ge«i
Qflittih} Iqp Ibe f$K4wglf^ HX Is thca^oie «i«^ttred bjr the cy*
liMlev Ihtl Konlil b^ dMmhed by the xetUiBf^ £S i-evohang
abaulGBi ^ipb QrUnidfer is triply Qfibecoae £^. Thus it
qqieim^ th^t ^ tibe thii^ solids whicb takea together aie e<|iiAl
ta the firustom HhcC (the in^enent of the cone AHA that is
pradvoed wbik A£ is iiicce«9ed by £B)^ the cjUader HAfZ
jneaapures the first fliuLion of the coae AHA> the hoUow solid
dcsoribed by the ndapgle HX measures one half pf its se«
QQttd Au^iqUj and the cone. £1^ measures pi^e sixth past of its
l^ufd iunkm, the axis Iiei^g supposed to flow unifonnijr, and
its flttxiofi being veproaenSed by D6.

134^ iM the pcMBti^ describe the right line l^insuoh % mmt^
ner that em upright parallelppipedoQ upon a gtven iiwariahle
hue, as the squane of any giyen line fi, ^nd of a height eqni|)
to Vp, iD^y be almfgns equid to the pyramid APMei (fig« S^,
aatfl^ ih^ci^otaoH of ji being then siaiilar to that with whidi the
pjnsymid flows> the acceleration 9f its motion sholl increase uni*
fimsdy^ <ir4he iffpssnieatsof its Telocity generated in aay equal
times that succeed after each other shall always increase uni*
fiu«^> wfaoi ^e aoljon (rf* the point P is iipifisni. LstdS
h»d9aaitK6libj p iB0ny giyien time ; Jet i/C he t|ie space thai
would be described in 4hat time by it with its modoii conti^
i^ued laii^insily £nass d; and Igt 4fht the space that would
hflne been deacrihed if^ theiuo^ time^ if the aooderation of its
HMitiOnhed been ioonttnned unifawniy ftom that tern« Then

T^ 9 f f 4 C/S w /

%

ft ^

i

a

•

z

t T

t

9

> ' ^J f .J

a

sh^BaprighipataUfliopipedotsontiieipyaiial^base thalts fup*
poead e^pal io 4he aqoaie of e^ and of thealtitades dC,4^ and
/S,he Mspcdivefy equdto the prisms EBeIGt^EHl€Ni/uKl4he
pjmsoidi^Nit; and^heri^lioesdC, C/.and/BshaD.measiwe

K3 the

150 Qfihird'^Nttxiens. B<k)ltL

the.fiiBt fluxion' of ¥d, one half of its second fivoAoUr and cue
sixth part of its third flusion^ respectively «
- 1d5u Let the points q and r detik^ribe the right lines Xxand
4^2 in sueh a manner^ th&t Xq may be always equal to the space
that would be described by /^^ and Zr may be equalto the space
ih^ would" be described by qy if their motions wei^ continued
uhifbilnly ifrom the terms p and q, in the same time in which it
iksii{)t>osedtfaat p would describe dC with its motion continued
ifiiformly from d. ThQn the motion of; shall be uniformly ac-
celerated^ and. the motion of r uniform> when the motion of P-
ioi the Ime AG ffig. 36) is supposed uniform. Let dS^ Vv, and
lUke spaces described by the points p, q, and r in the same time;
and let VI be the space that would be described by q in tliat
time by its motion continued uniformly frctoi V. Then dC, \l,
ahd T/ measure the firsts second^ and liiird fluxions of Etf^ re«
ipectivieiyi Theuprighjbparallelopipedonson the square of the
id/^ariable line'a^ of altitudes respectively equal to dC, Yl, and
T4 are equal to the solids EDelGb'^ NHEe i^nd NALe; C/isone
hai£ of VI, sndJB is one sixth part of Tt ; so that. dS is equd
1ot|ie sum Q{.dC,x>ne half of V/, and erne sixth part of T^^talceu'
together.: '•, '•'..•

r .156. BeeouseySis the differenceof cS and df, and is equal
fo.bnesisfth part of T/ ; it follows^ that one sixth part pf l^e
tfaurd fluxion raf the fluent Fp is measured, at any term, by the
Aiflbvetlce ofdie inotement^ that iaire produced in a given time^
whto'the aotelemtion of the gena:atingi motion is continued
unifd^-mly, and. .when the incoease of this acceleration is con-
tinued uniformly, from that term ; and it is determined by a
^i fliila r difference when the generating motion is continually
retarded, or when its acceleration decreases,

137. The rest remaining as in* the 129th article, \etBD(Jig.
37) be equaLtoDG, and the plane OBo parallel to £De meet
AE, Acy EI, eiy and eNin the points O, o, Y, y,^ audit. Then,
the frustum RDeOBo being equal to the excess of the sum of
tl^e prlsmEDeYBy and pyramid cc^.ebpve the prism EYOi^,
tt< follows,- that the difference betwieen the frustums ED^HGA
#rfd;£D<:OBoisdoutileof .thepiismEIHe«N,and therefore mea-
9iires tbei Beyond fluxion of the pyraloid ADEr^when ADfiowa

. /• unifoimly

ChAp. IV. Of third Fhxiom. 151

uniformly^ and its fluxion is represented by D6. It is als<^ ma-
nifest^ that the frustum HGAOBo exceeds the prism IGiYBy
by a solid that is equal to the sum of the equal pyramids eiNA
and eony ; and therefore the difference between that frustum
and prism measures one third part of the third fluxion of the

pyramid ADEe. * • ''-

138. In the same manner (fig. art. 1S4X if f^ aud dS be
described byp in equal times^ and dc be equal to dC, then the
difference between. c2S and di is equal to 2Cf, or to V/ ; and
the difference of fS and cC is double of fS, or equal to 09e
third part of T^ The former difference measures the second
.fluxion of Td, and the latter me^ures one third part of its
third fluxion^ when dC measures its. first fluxion. From which
another theorem may be deduced for determining the third
fluxions of quantities. If fi, dS^and Sm b^ described by the
.point JO in equal times that succeed after each other^ th^ It is
easy to see that T^ is equal to the excess of the difference be-
tween Sm and dS above the difference between dS and fd> lo
.other cases, the ratio of T^ to that excess, or of one third of
Ti to the. diffidence between fS and cC, may not b^ a ratio ojf
equality ; but^ it approaQhas to that ratio^ wbea those increr

. ments arc contumally diminished, as its limit.

139. It appears, from what was sbowp in the introouctioi^
that the third fluxion of any solid that can be generated by a

, conic section revolving on its axis is invariable, wh^n the axis
, flows uniformly ; the parabolic conoid excepted, which hasl^
second fluxion constant, and has no third fluxion. This ao-
count of third fluxions i? similar to that we gave of second
fluxionfy in art. 97. We have deduced those illustrations of
them from the common geometry, because what relates to
second and third fluxions has been represented as veir
abstruse, and beeause it is the application of this part of the
method thathas been fomid to be most liable to mistakes*

K4 CHAP.

I5t OfthcFluiiMitfCkaMiHes BtfricI

CHAP. t.

Ctfthc tlutiom tf Quantities that arc in a eonftn^ed geamc'*
itiekl FrogreUio^ tbcjirst Term of whitk m imariabU.

lii yight m^es in A; ^tifl, AS^ AP, and AL bdtkg in gebmetrical
l^rdgtfeMott, let SA be taken v^^bt KB, A? upoh AE, and At
l^oh A/; ahd SPt sh^ be atWay& a right Mgle. Let SA be
invariable \ and^ while the point P with an nmform itiotioA
di^cribes any eqti^ Unes pP^ Vp, let the point L describe IL
atid L/ ; produce \p and ^ tin they meet LP produced in d
tod D. "nen, because the angte ^Vd is equal co Spd; the
attj^e PSp is ei^ul to LA: %xi, titnce SPL is b right aft^e,
ttle atv^ SPp is equal to HH. D^erefote the triwgted I^^
leA i&re simllat, Ih like manh^r, the triangles t^, LD/ i^tB
ridiH^f. thde^oi^L/ist6Pjpttsl)Li5tt)SP,M4liiit6pp
a^^ is to dp; ^ that L/ is to Lias DL is to ^ But the
angle PSD is i^qual to 3Dpe, ox j)SA, and the angle PSrf is
^quai t6Pp(f, 6t pSA ; and consequently, the angle PSD ex-
'c^6 t^ bV t>d^, edual to pSp : so i9iat DL being greater
%ta'dt, U U gre&tcir ft^h LI. iTierefote, when the motion
dfPts unifdrm, the s{)aces described by Im any equal succeed*
Ing ti^es^ ]^rpetually increase, and \X& motion is accelerated.
141. Because the ancl6j)Sl) Is equfil to ;)PD, or PSA, it is
Ife^s than j?S/, and;?!) is less t'han pt Therefore T^th less than
«pf. I^t thfe angl^ pSrf being equal to dPp, or PSA, it is
\i^?LtA than pSl ; and, ^ beihg ^erefote greater l!han pi, dX

r42/lTie velocity 6fLls to the velocity of Pas SAP is to

SA. For, if the ratio of the velocity of L to the velocity of
P>be a greater ratio than that of 2AP to SA, let it be the same
as that of 2Ap (any quantity greater than SAP) to SA, or of
^l to Sp. Then, because D/ is less than 2p/, the velocity of
h shall be to the velocity of P in a greater ratio than D/isto

Sp,

QbipuT. imn g€milrical Pr&grt$d<mi 15$

Spf or (the triangles %»?, D/L being similar) li is to Vp. But,
the motion of L being acceleratefl^ while the motion of P is
nniform^ it follows^ from the first couom^ that Id is greaterthan
the qpace which would be described by L^ if i^ motion was
cotttiiiiied luttinnnly while P descdbcs Pji^ TlMefism the Te-
iMity pf L is to the velocity of P in ^ less ratb than U is to
Pjp. And th«»e bftk^g coatjradictDifyj it foUoviB, that the vdo-
citj of L is not to the velocity of P in a ratio greater than that
of£APtoSA« Let the ratio of those velocities be less than
ihat ^( ftAP to SA» <^ be the same as that of £Ap tg SA^ or
of^ItoS|»^ tvhioh bemg leal than diatof AtoS|>(becaitse
41 k gveater than "SplX ttte vdocity of L shall be to the vel»-
cilyof P in a less talio thdn that of LI to I^. Bolt it foHows, .
&cm ilie seooiid astkiiby that U ia to Pp lA It less ratio tha^
Mlocil^ of L is to ihe velooity of P. And these b^ng contra*
^k:bairfi,it ay^peocs, that the ^ocity of L is to the velocity of
P^ or the fnidon^f AL to ^e fluxion of AP^ as 2APas toSA.
143. Ijettlie«nglesSPL(/g.39)J'Uf »LM[N,MNR^.beal-
ways rights and the angokr points P, U M, N, R, S^c. descnhe
always the light liAesAE, A/; Ae.AF^AE,^. TheashdlSA,
AJP^ AL, AMj AN, AR, Ife. be abvays a series of quantities in
A ootttimiedgeometricalprogitession, ithe&Btteimof wfaichSAis
iliirnahle. While the pdntP describes any eiqital linespP,]^
t»n the line AE, let the points L» M, N, R, 4^. describe the
^spaces IL and U, mM and Mn^nN and Nn, andirR and Rr,Sfc.
on the lines Af, Ae, AF, AE, the angles Spl and Spl, plm and
^bn, bun and tmn being ahvays right. Jjdt Ip and Ip prodaced
«ieet LP prodnced in d and I> ; let asl lodm/ prodaced meet
^L ingauid O ; let as and nm, m and rmfifCfttoAoced meet
NM, RS, 4rt;. in h and H, ft and K, ^. respectively. Then
it ismanifeM» Aat the triangles PSp, hHyMga^, Nin, ^c, and
4he tristnii^ P%t, LD4 MOm, NHn, RKr. i^c. are each a
)serie84)f ssaiilar triangles. It is also evident, that the angles
f&A, PSD, LD6, M6H, NHK, ifc. and 4be angles pSA, fSi,
mg, M6A, NH/^ ifc. are eaeh a series of equal angles ; and
^tat'anjr angle of ihe former aeries exceeds always thecorres-
^poadsDg aai^leof ihe latter series by the difference pSp.

LEMMA

1^4 Of the Buxiom of (Entities . Book I.

t i

' LEMMA .VI.

144. ^fVhen the first term ofn geomttrieal progression, is invarta-*
He, and the second term increases uniformly^ each of thejhi--
• hzoing terms increases with an accelerated motion.

, Because the triangles PSp, Ldl, Mgm, Iff An, R/ir, iic, are
similar, the incrementa pP, IL, mM, nN, rS, i^cJ are in Ae
stfme proportion as the right lines SP/ rfL, gM, AN, AR, ^c.

-aid, the triangles PSp, LD/, MGm, NEDi, 4rc. being also simi-
lar, the increments l?p, U, Mm, N», Rr, 4r^. are in the same
pibporiion as the right liAes SP, DL, GM, HN, KR, 8^c. There-
fate, pP being ec^aal to Pp, IL is to L/ as dhk to DL; mM
istoMm^ss^M istoGM, nN istoNnasANistoHN, rR is UrRr

. as AR is to KR. ' But DL is greater' than dL, GM is greater
than ;M, HN is greats than AN, KR is greater tha;n AR,
and so on ; Because the angles PSD, LDG, MGH, NHK, S^e.
exceed tit* angles PSirf, LDg, MGA, NHA, ^c. respectively (the

I excess being always equal to the angle'jpSp). Therefore U,
M«i,' Nw, Rr, 4rc. exceed LI, Mm, No, Rr, '8^e. respectively ;

' and, the motion of the point P be^ng. uniform (so that the
equal lines pP, P/) may be described by it in equal times), the
motioilsof 'die points L, M, N, R, i^c. are. perpetually accele-

mted.

145. It is manifest, that the lines D/, Gm, Hn, Kr, Sfc. nx^

V respectively less than Qpl, Slm^ 4mn, 6nr, S^c. but that the lines

di, gin. An, Ar^ ift, are respectively greater than fijd, Sim, 4m n,

5nr, ifc. For the angles pSD, IDG, mGU, nHK, ^c. being

always equal to each other and to the angle PSA, and the

• angles j^S/, Ipm, mln, nmr, ^c. being also equal, the latter air

ways exceed the former by the angle PSp. i Therefore as pi

is greater than pj), and D/ therefore less than 9pl ; so 6^ is less

than £//», and iim less than Sim ; Km is less than Smn, and Hi?

less than4mn,* K/iislessthan4iir,andKrislessthan5iir. intl:fe

same manner, the angles ^Sd, Idg, tagh, nAA, Sfc. being alwajs

equal to PSA, and the angles pSl, 1pm, mln, nmr, S^Cf being

* e(|ual.

PUfe»'Jta/:*<i.»W.A

* N

I

Chdp. V« in a geometficat ProgremM* 155

equal> the former angled always exceed the latter by the angle
PSp. Therefore^ as pcf is greater than pl^ and consequently <fl
J8 gl^ater thani2pl ; so Ig is greateir than Slm^ and gm greater
than Sim ; mA is greater than Smn, and nA greater than4nm ;
nk » gyeales than 4nr, and kt greater than 5nr ; and so on^

PROP. VIII.

146. The fluxion ofamf term AN of a geometrical progreM^n^
• the first term of which is invariable, is to the fluxion of the
' tee&nd term AP in a ratio compoundid of the ratio of those
terms ani4he futio of the number of terms whichpreeedcAN
' to unit.

The fluxion of any term AN is to the fluxion of the second
term AP in a ratio compounded of the ratio of AN to AP^ and
of the ratio of the number of teiins that precede AN to unit ;
that is^ in this example^ the fluxion of AN is to the fluxion of
AP as 4AN is to AP.4I First, let the motion of P be uniform^
and the motion of N shall be accelerated,' by lemma 6. Then
if the ratio of thu' velocity of N to the velocity of P be not
that of 4AN to AP^ or of 4AM to SA, let it first be the same
as that of 4Am to SA, Am being any line greater than AM.
The triangles SAp, Amn being similar^ 4Am is to SA as 4mn is
to Sp ; but 4mn is greater than Hn (by art; 145)> and 4mn is
to Sp in a greater ratio than Hn is to Sp, or Nn to Pjp. There-
fore the ratio of tlie velocity of N to the velocity of P is great-
er than the ratio of Nn to F^« But the motion of N being ac-^
ceferated while the motion of P is uniform, it follows, from the
first axiom, that a less space than Nif would be described by the
motion of N continued unifotmly in the time Pp is described by
P. Therefore the velocity of N is to the velocity of P in a
less ratio than Nn is to Pjp. But these are contradictory, and
therefoie the ratio of the velocity of N to the velocity of P is
not greater than that of 4AM to SA^ or of 4AN to AP. Let
the ratio of those velocities be that of 4Am to SA, Am being
any line less than AM. Tl^cn^ 4Am being to SA as 4mn is to

1

Sp» and 4mm WUig km tbm Ap (bf ;»H» }43)|i H £4Imi6« tbui
the melocJty of N ia to ibe veloei^y ef P tp a l99«nklio thaji Aa
istoSp^orNnisto Pp»* 13itt» b; tfae peMil4 Aioodsi^ » graill^
space than Nft ivmld b» de9eribe4 by tkoinatidiiipf N eooiH
nned HaifoffmIjr>:m the time pP is 4eaarib^ bjr P i wA thmen

fore the velocity of N is to the Telocity of P in £^ greater ratio
than Nn is to Pp. But these being also contradictory^ the ve-
locity of N is to the velocity.of P |:)f i)^er in a greater nor in a
less ratio than that of 4AM to SA^ or of 4 AN to AP; and there*

fafetiKfloKiM pi AM is to theflwiw.^^P pAi^ly !» ibt
aBtme ratio «3 4AJK. i& to AP, AU th^ akber q9§^ of Uiis f^o-
posHioo^ whM the wc^tJM of P is vairi^^, 9tf, ^mly ditdM^
iii;Ai klm case;, hgr tbt Al^Yestibi geperp> i>]mfl¥^<» Aui H #«^b-

▼ious^ that this demonstration is applicable in the sajii^ ii|4n-
per td any other terms of the progression.

• t

PROP. XX.

V . • ' ' . '

• • • i -

147* IieJhxwmsqf4mffiwotcmnn.*§f(Hmfrk¥tprfigrm
Aijina term/fffBpiiAk mvumUi, axt^iu 4 ratia om^miBd^
tdrnftht ratio ofthoK term t9 emh 0iker, und of tit rmii^
wfiht fliiMikrs Uwt *ham hsua tmmy tanm ftre^eda thtm in
Ute pvAigrcsnon*

TMa fioBova ftom fth<i kaft prapositiifla. The jfluxioQ of A^^
fetwcampifv ia to <be ^qsioti of AM as :4AN is to d^M. For
the ftiiion of AN is to !tke Auiriopi ai AP as^ASf is to AP,
bjthiBladtpropQBiikA; «iad(tbeflj^daiiaf AMistotbsflajdoB
ef AP as sAM is to AP> bjr tlie.sane. » HiereiftKe the Samoa
oTAK isto dbe flntfion «f AMag ilAfil isioSAM^ and i^m
Vianifeat, liukUheaaaHi dflawnn^wtiaamay lie afpliadwhan the
fuooions of any odiher tesms of l^e pm^sabn axe isunpared
ts^iiter.

MS. It Mlowrs £rom svliat has ioeea idemonBtiwted, ^Jtmt,iS
Ae motiacis of tbe points P, dLi, M, V, fi> ijre. vese oonti^ued
nnifonnly, fines reipeotiffcly leqpalto.SA, ^AP^dAI^ >4AM^
^MH, %. woidd be (j^eaonbed by tfaem ia^ibe saaie ^ftime.

140. llie

M9. Thi^MMnd^ thinl, mM Higher JaacMoa of aiijr ttva of
IH« t>i-d^6^iiM>«Miy beite|prdteii«dlbjr ooiilttB imiltiplMof tbc
^r«<M!ffi«ig tM[i!i whea tbe first Btixbuvf the MCted temi AP
is coxistail^t)lr thi iHOtion of P is iMMfeda>> and is fepccsBilBl
by the invariable right line SA. Fpr the velocity of the point
N^ or the first fluxion of AN, being represented by 4AM ; the
fluxion of this velocity^ or the «eooBd fluxion of AN, may be
represented by four times the fluxion of AM (which is itself
measured by 3AL), or by 12AL ; the third fluxion of AN is
nfiMM^Bi by viAP (1»ec«uM the fluxion «f AL as nfSMeiAed
by 9AP), its fourth fluxion by 24SA,' and it has no fifth fluxion.
150. If Att be a mean^roportional between AL and ikf (^g^
^, lUid AP 9ow tttnfmttJyjiiexi iu SLk equal to the Hne-tfiift
tM^cM be ^en^ratied by tlie ttotion widi winch AL flows cond*-
tLuei turifottnly fdr *the time in nrhicli P describes Tp^ anfl "die
diJRfretiee )ictwi>kt hi and Lte is what is generated in eonse-
(({(letide t>f the %icce3eration of tiiis motion during Amtttme.
fill: Aais to AL in the siA>dtr|ffie«te ratio of Al to AL,t>ria
theititib'of^to AP; aa»ftLttistef5pas AL is to AP^wat
AP fe t6 SA. mret^orfe '2Lu is to 1^ as '«AP is to BA, w as
Ihfe i^oefty of flife point L is to the vetecfty of P. Let A%und
Aft ht two mtpim proportronttls "between AM and Am ; mid if
itit tnotion of M ^<ras conftfaiued tftrffprmly white P describes
l^/nlhie #ditld be described by it tqaA to SM*; tf (lie ac-
celeration of that motion was continued uniformly from tlie
ttune term and for the same time# ti line would be described
by it equal to Shk\ and what would be generated in consequence
of this uniform acceferation is equal to thrice the diflerence
1>6twixt A/: and' MA. The differenciel>e^wikt Mm and3A&, or
the excess of the difference betwixt km and hk above the dif-
ference betwixt hk and MA, . is what is generated in. conse-
qilence oTthe increase o^f that acceleration during Ihe same
lime. 'iTie proportions betwijrt those differences and the lines
4liat latMtBme the-fiist/secoiid^ and\thiid.fluMions.ofAM7 easily
4ppi6ar, tmd 'hsvi^ been ladready showh in the hot i«haptef .
'Any term haJs Huxions dfastnany degrees iats thei'care terms
thfl^ precedeitin ihe .profession, when the ^xion of AP is
invariable ; and the increment of any '^esfia 4bttt.i8'g«ierBted

in

158 Of Logarithms; and the Fluxions Book I*

in a given time may be resolved in this manner into as many
parts as it has fluxions of different orders ; and each part may
be Gonceived W\>e generated in consequence of its respective
ftixion. But this will appear more easily aftearwards.

CHAP. VI.
Of Logarithms^ and the Fluxions of logarithmic Quantiticsm

151. JLHE Logarithms were invented for facilitating com-''
putations in Arithmetic and Trigonometry, by the celebrated
Lord Napier, Baron of Merchiston, and have been found
since to be of great use in the higher Geometry, particularly
in the method of Fluxions. Their nature and genesis is pro-
posed by the inventor in a method similar to that which is ap-
plied in this doctrine for explaining the genesis of quantities
of allsorts, and is described by him almost in the same terms.
He begins his treatise on this subject, by defining that a line in*
creases equally, when the point that describes it moves over
equal spaces in equal times. Let A (says he*) be the term
from which the line is to be described by the flux or motion of

A C D E F B

n

c d e f mg o Q R S

the point P. Let it flow from A to C in the first moment (or
in any small part of the time) from C to D in the second

* SifpuncXui A k quo Aiceitia sit lineajlwnt alierius fiimcii qui tit P. flitat
ergo priwto momento V ah KmC^ $eeundo wuwiemta 4 C m D, &c. Mirif, Logir.
Canon, deicript. defin. 1. He afttrwardt lays down a postulate, similar to what we
assumed in the first chapter, in these words : Quam quoUbet moiu \S iordior ^S
wlodtr daripossil^ sequefur necessarid cwque wwtui ^quheloeem (quern net tarJiorem
nee -veladorem JefiniwmO darip09$e,

moment.

Ch^. VL of logarithmic Qianttties. lSf§

monaeDty firooi J) to £ in the Jtbird ; md so on for ever^ de*
scribiDg always the equal parta AC^ GD^ D£^ £P^ t^c. in equal
times, N Ttiis line is then said to increase equally^

152. By his second d^nition/aline decnaaes praportiet^
mUy, when the point that nioyies.Qver.it describes sach parts in
equal times as are alwajis.in the same oonstant ratio to the
lines from which they are subducted^ tir to the distances of
that point at the beginning of those times from a given term in
the line. Let the ratio of QR to QS be any given ratio ;'let
ac be to ao, cd to co, de to do, ef to to,fg to fo, 8^c. always in
the same invariable ratio of QR to QS. Suppose that the point
p sets out from a^ describing iic^ cd, de, tf,fg, 4rc. in equal parts
of the time ; and let the ^pace described hyp in any given time
be always in the same ratio to the distance of p from o at the
beginning of that time. Then the right line|K^ is said to decrease
proportionally. It is manifest^ that the lines, ao, co, d^, c6yfot
i^c. or the distances of the point jp from o at equal succeeding
intervals hi time^ are in a continued geometrical progression.

153. Suppose now that the uniform > motion of the point JP
in describing the line AB is equal to the motion with whiehjp
sets out from a in describing the lij^e^a; and the line APthai
is described by P with this uniform motion^ in the. same time
that oa by decreasing proportionally, becomes eqifal. to. op^ is
the Logarithm of cjp.f Thus AC, AD, AJE^ AF, ^c. are the
logarithms of oc, od, ot, of, Sfc, respectively; and oa is the
quanUty whose logarithm is supposeid equid to not|iing«

154. In like manner, suppose the line oa to increase propor-;
tionally; that is, let the point p move in ^he line oa procfaioed
beyond o, and in any equal times describe spaces prc^rtional
to its distances from o at the beginning of each time; so that

."i ...

* Limea ^pprHcnaltier in ireviorgm dccreicere.Sictt»j quum punciw earn fmm-t
currens ^tqualihus wiomeniis segmtnta ahtciadit cjutdem eoniimio roHoms oditBeps^m
quihus ahscinduniur. Ibid, defin. S. ' . ^ *

t Logariihmus cujtisqut snA% est maherui qu^ ^rcxim iefiniens liriram qtuB
ggquaiiitr crevii^ inlerga dim sdi^ UHus UnecK^apotttumaUter in ittutn stnmn de-
crevitf exiaienle utroque moiu $^hrano, atqu$ imijo ^pdwloec^ Ibid* That tbis
doctrine may be more general, we abstract from nuiAben^ asul repi^aient the logafitbmt
by Sues, as weU as the quantities of which they are the Ipg-arithms. . •

o

a

the

t69 CffJUganam,ttmitheBa)ei<m Botlil.

llie im»eiiifinlaar^cii»ife^^iirc.ttMiybeie9eiibedbyitin<N)Dal
timesj wbeti flchtoa0,edtAC0,detodo,efibee,lfc, wiwajB
in the %ame iimuaabk ratio. Whenp teto cmt from Oj let P
aetovtftmD A mthiiiie<|lial vdocity; then^ the mpiioB of P
fcek^ eontoiiied iini£armfy while &p iacreaaes pFoportionallj;
the line AP if alw jiys llie togarithm of op ; and the lines AC,
A]># A£^ AF, S^c. ate the iogarilhma ^^, ad, tff^o^^rc.fe-

a m d €

i«p

P

A € ]> e F G

W»^—i ^P— ^— ^^ "^F— I^Wi^-^

ipectively. It is manifesty that the Knes oa, oc, od, ee, of, e^c.
aie in eontiimed geometrical proportion. If ac ^^fg be de-
aciibed byp in equal inter? als of time, axid an,fin be described
hy it in any etiul paits of Aose times \ tlien shall ac be \ofg,
and imto^,«S6€iistoof^ or iLS oc istoog*, or as on i3 topm;
tfaat is, the spaces described iy p in equal times are in the same
fvopoition to each other as ^ distances of jp from o, at the
keghming> end, orany sim9ar teim of those times.

1$6. When n ratio is given, the point p describes th^ dif-
ferenee of the teitns of llse rstio in the same time. TTIiei) a rar
tio is duplicate of anoAer ratio, the point p describes the dif-
ference {rf'ihe terms in a double time. When a ratio is tripli-
eate of anc^er, it describes the difference of the tenos inj^.
triple time v ttd so on. Thus the ratio of oi to on is duplicate
ef the ratio of oc to oa ; and, liie lines ac, cif being described
\jf in «qttal times, die time in vliich ad is described by it
is double of that in whidi it describes ac. The ratio of ot to
oaJatEipIifiila 0f the Mtio ofoc4o«a, and the time in m^xtAx
«eas<d«eribed bf jp istrtpAe df that in ^^ch it describes ac.
In the same maxmer it a^pearsu that; ^liea a jatio 13 pcm(»g(|iid*-
ed oftwj»armQie gatiaa, Ibe paiot ji^dMriiben the diffMPepce
fifthetcmsoftlHiftTaftio, isa -a timeeqmd to the sum of llie
thnesanin^fatdi it describes tbt difieienqes of the teans of the
sim;^ ratios of whidi it is compounded.

156. WhA

1

lS9i Wkal «t ham rad of the times «f the motion ofp
wbcn 0p ia^i€Mef propostioQaUy^ is to fae«ppiied io theq^«Q4ii
duift eMdtetoribed by P in tho« time* with iti nnifona ouHieai
imd htnoe the diief piopertieB of the logarithms are dedui^ed*
The 4i9Btemo9 4)f the logarithns of the terms of the sraie nulia
is ahmfs the Sine. WhenaratioisdiipUcAteof aootherratie^
4ie iSffcMAoe of tiie logaiitfams of the terms is douUe ; wbea
a mito is tripKoato, thai' diffntnoe b triple ; and so on,. The
di fl b r e n eeoflfcetegarithmsofthetegmaofacomLpPiindr^tio ii
equal to the sum that is produeed by adding together the dif-
ferences of the logarithms of die terms of the simple ratios
fiwi which H is iumipoanded.

157. Thus logarithms are the measures of ratios. The ex*
cess of the logarithm of the antecedent above the logarithm of
the consequent measures the ratio of those terms. The measure
Of the ratio of « greater quantity to a lesser is positive^ as diis
ratio cQflipsKmded wkh any other ratio incxeases it. The ratio
of equality compounded with ai^ othe mXio oeitiier increnfiei
nor diminishes it ; mdits measure js nothing. The measure
of the ratio of a lesser qnaatit]; to a greater is negative as this
n^io compounded widi any other ratio dimtniAes it The xat
lie of any quantity A to unit compoanded with the ratio of
unit to A produces the ratio of Ato A^or the tatloof eqeafity;
and the measures of those two ratios destroy each other when
eMediogether; so that when the one is considered as positive^
the other is to be considered as negative. By supposing the lo^
gartthms of quantities greater than oa (which is supposed to re*
piesent unit) to be positive^ smd the logarithms of quantities
less than it to be eegative^ ite same eoks ser^for the open^
tfoos by kgaritlttiB^ whedler ibe quantities be greater or Je^i
tfMtiee. FiomwhatiN^hevetfBid^itiseeqrteaeehDwtbele^t
gariflnas serve far abridging oomputations. Because uoit is to
my qaantity Aas Bis to theptodact of A and B, and the W«
garifkm ofuidtisnotldag^ihe logaridunof cgBgrprodnet is
equal to-flie sttm ef the logmMsins of the factors, and the Ie«
gioMhm of aey power of A is to fhe fefsotiim of Aea the e«r
peeem oPthet peeisr is to «mit.
VOL; L L 168. Wheft

I^ Of LogArithms, and the Iluxiom Bool L

158* When op increases proportionally, the motion of p is
pferpetually accelerated ; for the spaces uc, cd, de, SfC. thai are
described by it in any equal times that continnalij soooeed after
each other^ perpetually increase in the same proportion as the
Hnes oa, oc, od, ^c. When the point/> moves from a toiraids o,
as in the 152d article^ and op decreases proportionidly, the mp«
tion o(p is perpetually retarded ; for the spaces described hy
It in any equal times that continually succeed after each other^,
decrease in this case in (tie same proportion as op decreases*

PROP. X.

159. The fluxion of a quantity that increases or decreases pro-?
. portionally, varies always in the same ratio as the quantity
. itself.

Let the line op increase or decrease proportionally, and the
Telocity of the point ji (or the fluxion of op) Aall always vaiy
in the same ratio as its distance from o. Thus the velocity of
p at the term c shall be to its velocity at any other term g, as
oc is to og. For, if the ratio of the velocity of p at g to its ve-
locity at c be greater than that of og to 9C, let it be the same as
that of any line ok greater than og to oc ; and let oA be to oc as
og is to ok. Then is gh to kc as ok is to oc, or as og is to oA >

and the lines

q n k c g k ke, gk are de-

p scribed by

the point p
in the same time, by the supposition* Let op first increase pro-
portionally ; and, because the motion of p is then perpetually
accelerated (art. 15B), it follows, from the first axiom, that a
less line than gk would be described by it, if its motion waa
eontmued uniformly firom g, in the same time that it describes
g4 while op increases pr<4>ortion£dly ; and it follows, from the.
•eoond axiom, that a greater Une than kc would be described in
an (Bqual ttmebyp,if its motion was continued uniforixilyfrom
the teiia v^ Therefore the velocity of p at g is to its velocity
at c in a ratio that is less than, the ratio.of^A to kc,, or of o& to
9C« But it was supposed; that Ate velocity of p at g i; jto.fte ve-
locity

Chdp. VI. ofhgaHthmic QMntitics. 16S

locity ofp at e as oh is to oc. And these being coDtradictorj^
it fdlow8> (hat the velocity of jp at g is npt to its velocity at c
in any ratio greater than that of og to oe. In the same manner
itappeaiSj that the velocity ofp at c is not to its velocity at
g in any ratio greater than that of oc to og ; from which it
foikiws, that the velocity ofp atg is not to the velocity of p
«t c in any ratio that is kss than the ratio of og to oc. There-
fere the velocity ofp atg b to the velocity ofp at c precisely
as og is to oc ; and the fluxion of og is to the fluxion of oc in
the same ratio. When op decreases proportionally^ the de-
monstration is deduced in the same manner from the third and
Iburth axioms.

160. But it may he of use for illustrating the account of lo-
garithms that.is proposed by the inventor^ to demonstrate the
converse of this proposition from the axiomsj^ and show that
when the velocity of jp is always as the distance op, then this
line increases or decreases in the manner that is supposedbyhim.
Let the velocity ofp therefore increase always in the same pro-
portion as op increases ; let afhe to gm as oa is to og : and tlie
spaces afy gm^ shall be described by the point p in equal times ;
that is^ op sh{ill increase proportionally. For^ if the time in
which o^ is described by p be supposed to be greater than the
time in which gm is described by it^ let on be greater than oa
in the same ratio* J^et oa, oc, od, oc, 0/^ be a series of lines iii
contimied proportion^ the number of proportionals between oa
and o/'being increased till the term oc (which is next to od) be
eome less than on. Because oa is to of as og is to am, if og,oh,

o a en d e f g h k I w*

P

ok, olj om, be also in continued proportion^ and the number of
terms be the same as in the other series^ tiie lines ac, cd, dc, cf^
gh, hk, klj /m, shall be in the same proportion to each other re**'
spectively as the lines oa, oc, od, oe, og, oh, ok, ol, or as the lines
oc, od, oe, of, oh, ok, ol, om. Suppose first the line ac to be de-
scribed by an uniform motion equal to tliat ofp at the term a,
cd to be described by an uniform motion equal to that ofp at

L2 the

1 64 Of L&garkkms, and ih ttmans &o6k L

the term c; and ek, ef,gk, kk, My /m^ tobedeseribedby uniform
motions equal to thoM of thepoinlp at the respedtife termsif^ #>
g, k, k, t. Tben^ these spaces Me,cdy Jk, if, gh, Mr, hi, Im beiog in
the same ratios to each other a| the irdooities of tli# tmtfenn itto*
ttons by which thay are Supposed to be dosoftbedjit fbll^in^ thait
they are all described in this case in equal ti«ias; and, the t&iiti
in which a/*knd gfft are thus described behMr Mttitiiiilliiiles ctf
the times in which ac and gh aro described) tHey arethev^ra
equal. Suppose now the same spaces B€, ed, dd, if,gk, kk^ kl, Im,
to be described by uniform motions pespacti vely equal to tihose
ofp at the terms c, d, e,f, k, k, /> m, and the tiiaaa iin whiek
they are in this case described are all equal to each otlier, fbr
the same reason ; and^ consequently^ the times ia whiok d^and
gm are thus described are equal in this ca«e ajso. The time tji
which af is described in the frst of tliose two casesy is to the
time in which it isdeseribed in the seoend^ as thetime ia which
the line ac is deseribed in the first is to the time in whioh the
same line is described in the second (Eiem> 15. b), or as the
velocity of the uniform motion by which it is described in the
second is to the velocity of the uniform motion by whMi it is
described in the first ease ; that is^ as oc is to (mi» Ficm which
it follows, that the time in which ^?^i« described in the first oase
is to the time in which gm is described in the second (which is
equal to the time in which of is described in the smne case) in
the same ratio of oc to oa. But the motion of |i being acoeie«*
rated continually while op inoi^eases, the time in which ac is de*
scribed hjp is less than the time in which ac is described by an
uniform motion equal to that of j9 at a accordmg to the first
supposition. In the same manner; the timesin which cd, de, and
e/are described by p are less than the times in which those
spaces aredescribedbyuniformmotions respectively equal to the
motions of p at the terms c, d, and e ; and, consequently, afis
described by ji in a time less than that in whidi af is described
hi the first of the two caaesthat were supposed by us* In the
same manner, the time in which gh h described hyp is greater
than that in which gh is described by an uniform motion equal
to that of p at A ; and the time in which gm is described by p
is greater than the time in whioh gm is described in the second

of

of diott bm ^atifi. Tberefbi^ ^ time in ^ioh n/is ddscrib'^
#d hf p « to the. ibse in irliieh gr^ n described by it^ in ft
kfft futio ibim AetifM m which of it desottbed in the formcsf
case is to the time in which gm is described ia die latter ; thttt
isj in ulesi hiti^ tblti o^ is to 9a ; and, eoii9eqiieiitl7> ifi a ksi
Yatio than ofs it Co oa. But the times in which af aod gm sura
descvibed by p were vetyposed to be in the same ratio as m is t^
*a. And theat being oontradictnry^ it follows^ that the tiaoe
in which o/* is described bj j? is not greater than the time in
which gm is deacribed by it* In the same manner it apfNsars>
that the laibler of those timaB cannot exceed the foriaen
Therefore the lines afand gw$ are described by p in equal tim^A
when the velacitjr of p Increases in the same proportion as op
increases. A dcmonstratiesi of the same kind is easily adapted
to the case .when op is sapposed to decrease.

Idl* Cor, I. The fluxion of any quantity op is to the dax--
Urn of its logarithm AP (fig. art. I5l and 154) as op is to oa,
which represents nnit, and has its logarithm equal to nothing.
l%e motioiiof F>hy which AP> the logarithm of op is describe
ed^was sappctaadChs art. l^Sand 154) uniform and equal to the
telocity of ji at ie^ oa being the quantity whose logarithm is
nothing. Thcrefiiifi, by tfail prcq>ositaoin; the velodty o(p at
any term of the time is to the constant velocity of P as 0/1 is
to oa. When the logarithm AP is supposed to flow with a
variable motion^ op does not increase proportionally : but^
from what we have shown^ it may be demonstrated by the
eleventh general theorem^ that the velocity of p is still to the
^^^lacily 0S P (op being always the line of whieh AP is tb^
logarithm) as op is to oa.

16s. Cer. II. When op increases proporUonally^ the incre-
ments that are generated in any equal times are accdrately in
the same ratio as the velocities ofp, or the fluxions of op, at
the beginnings end^ or at any similar terms of those times.
Tbxm af it to gm (fig« art, idO) as oa ia to o^^ or as o/* is to
om, or as o€ ia to oA. z

l6d. Car. III. When op increases or decreases proportional-*
ly, the fluxions of this line of all the higher ordersincrease or
dtcaoase in tfaesaaaeproportion as the hue itself increases or de-*

L 3 creases

1

166 Of Logaritkms, and the FluxioM Book't.

creases ; so that one rule serves for comparing togethtt those
of any kind at different terms of the time. For^ the" velocity
ofp being always as op, it flows in the same manner as op flows,
and its fluxion (or the second fluxion of op) varies in the same
manner as the first fluxion of op, or as the fluent opitself varies.
In the same manner, when the second fluxion of op is consider-
ed as a flowing quantity, its fluxion (or the third fluxion of op)
is as the velocity ofp, or as the line op. It is evident, that^
the higher fluxions of op vary in the same manner ; and that in
this case we never arrive at any constant or invariable fluxion.
In treating of those higher orders of fluxions, we never suppose
any fluxion of op to be the veloci^ of a velocity ; but we find
that the fluxion of any order flows in the same' manner as op
flows ; and, considering that fluxion as a flowing quantity, its
fluxion is as the velocity ofp, or as the line op. Tiie fiiBC
fluxion of any fluent is not the velocity of that fluent, but the
velocity of the motion by which it is conceived to be generat-
ed ; and, in like manner, the second fluxion o>f that fluent iff
not the velocity of the velocity of this fluent, but the velocity
of the motion by which the quantity is generated that always
represents its first fluxion : but of this weaaidenough above, in
the first chapter, from art. 70 to the end, and in art. 07^ 134,
135, 137> and 138.

PROP. XL .

r

l65. Let the logarithms of two quantities he aboays ta each

other in any invariable ratioy andthefluxiom of those quan^

. Jit its s/uill be in a ratio that is compounded of the ratio of

. the quantities thenuelves, and of the invariable ratio of their

logarithms.

Let the points Q and P set out from A together, and describe
AB with uniform motions ; and let the velocity of Q be to the
velocity of P in the invariable ratio of £ to F. Let the .points
f andp set out at the same time from a in the right ]ine ab with
vek)citics recipectively equal to the velocities of Q and P. Then,

if

Chap.TI. of logarithmic Quat^itks. iQj-

if Of and op inoettie proportionally^ AQ shall be thu log^nthqii
of oq, and AP the logarithm of op ; and;, AQ being to AP aai
the velocity of Q i» te> the velocity of P, or as £ is to Fj iti
Mldws^ that thebgatdtbxn of oq is always to the lof anthxn of

d

a

e

k

., *

P '

9

1

A

C

•

K

B ]

B F

p a

ap in the invariable ratio of £ to F. When p comes to c, lei
T, q, andQ come to C^ k, and K respectively ; and AK shall be
to AC as E is to F* The velocity of 9 at A; is to the velocity
of ^ ataas o& is to oa^ byprop. 10, The velocity of ^ at a is
to the velocity oip at a as £Li to Fj by the supposition. The
velocHy of p at a is to the velocity ofp at c as oa is to oc, by^
firop. 10. Therefore^ by compounding those ratios together^
the velocity of f at & is to tlie velocity of p at c as the rectangle
contained by ok and £ is to the rectangle contained by oc an4r
F. But the fluxion of oA; is to the fluxion of oc as the velocity
of 9 at ifc is to the velocity of p at c ; that is^ in a ratio com-
pdonded of that of the lines ok and oc, and of the invariable^
aratio of their logarithms AK and AC.
• 166. Cor, I. When oa, ^ ^^ o& are in continued geometri<»
eid progression^ the logarithm of ok is always double of the lo*
garitbm of oc, by art. 156; and therefore the fluxion of oA: is to
the fluxion of oc as 2ok is to oc^ or as 2oc is to oa^ as has been
alrefKly demonstrated in the 96th and 142d articles. When ok
is any term of a continued geometrical progression of whic|i oa
and oc are the two first terms» and ok occupies always the same
place in this progression ; then^ because the logarithm pf ok itf
lo the logarithm of oc as the number of terms that precede ok
)a the progression is to unit^ the fluxion of ok is to the fluxion
of oc in a ratio compounded of the ratios of ok to oc and of t^
nnmber of terms that precede ok to unit^ as was demonstralM
\fk the eighth propositbn. In general^ when oc is any other temtN
ot tibe progvessioii^ the logarithm of ok is to the logarithm of oc

L4 as

1

tB» Of lAg4ritkm^ and tie fkxiom Botk Iw

»tke namhei ^i^tm ihk% imoed€ okiM to Ikt «nikv #f l««tti
fieU ^Moede (X) ; ahdthft fluxidil^fdJbiitothiflttjdQK of p^ift
iTaii(^^<ttapoMd6d of the tirtio of thoie minbari iiidof tlK 8^
lid#f <9A't^<>e^yfii^A^dlMK3li^traMlfii tht itiMh pmposliM
in a diiFerent manner.

? l Q7 > Cor. II. Xfit Qg4yyg^^ ptoportionally wliileopincreafl*
C8 ; let the motions of the points Q ind P be uniform/ but in
opposite directions from A, and be respectively equal to the mo-
tions o Xp sni 2 at th^ term a : then the velocity of q shall be
to the velocity of |) in a ratio compounded of the ratio of og to
op, and of the ratio of AQ to APj as before: but the fluxion of
o(j^tnast be considered inthiscaseasnegative^beeauseoj^dcieioai*

es. When op, M> aud 0f alft
o. " q i» p in continued proportion, AQ

Ihe logaritbitt of o^ ifl equal to
Q A P APthelog^^moffip^tliovo^

T ' •" bcity of f to to th« irtlodity of

jrasof tatoop; and any lines that measure the fluxions of tfao
terms oq and op are in the same ratio as tho^ tenoM Ae»g6lv«i«
lugeneral^when oq, oa vtniop, are terms of any geemeirieal pio«
gression^ and the number of terms from oa to oq (tndiiding ono
6f those terms themselves) is to the number ot terms froto o^
to op (including one of these) nmrniaton; then AQ into AF
as m is to n ; and the velocity of ^ is to the velocity of j^ in a
ratio compounded df the mtios of oq to op and of ivt to o. Any
quantities that measnre the fluxions of og and op are hi the saaio
ratio ; but the fluxion of oq is considefed as negative whtt the
Aixionofop is positive. Thus'itappearshowlhe^xioMof A#
terms of a geometrical progression aredetermined when any ia«
lermediate term of the progression is invariaMe.

168. Cor. III. In those cases^ the velocity of Q is to the v*^
lodty of P, or the logarkhm of o y to the logaiithm of dp^ hi «
ratio which cati be e:q)ressed by that of rational uniiihttB. Bol
the proposition is general^ and the ratio of thefluxioiif of 09 and!
op is assigned by it wheto the ratio of AQ to AP i# iavttriable^
thoug)i this ratio cannot be expressed by that of numberto tkwak>^
l^r. According to the notation trbtcb is now itt tise in Aigfkfias,
when AQ is tcrAPas m is to n^ and Mtwang Uttit^ ot iA eJfUia^
^ «d

•d bj Ai tad It ia iul^pMed («B id art, 165) tiiftt •f fcacDom

when ibepQintspMd f an oo the Mine aide of tlie point a» and

by A ^ when tliey are on opposite aides of a* Whetker iha

•Mfojom^X -^beratkmaliMrkfaftkmalt^ttiiethabaa^oMraalbt

iniiiOA of oft ia to that trktch ittendttred the fiatton of oe m a
fllllOGoa!i]^ifededi!jf the ratio of m teiii^ teoA oftheratioof o>to
t. This notatiori by Irktional eitponefits at>pea» fitst iti Sit
iJMMc IfmtM*^ letter to Mf . OldenbUtgh, dt OdblM 44,
l67d« If die ratio of AQ and AP^ the logarithni^ 6f t^ and o^^
be tarii^le, let it be the tome aa that 0f os' to M (t^r ifanlt))

one k

Q A P Q

aod>opbetngexpfeaaedby A>Ar by r,ef tauiy be expiealed bjr
A^(the loganthma of ponders aod theiff exponeata beiqg al«
waya in the aame rado), «Bd is called an iSponetUial qaaniity
ef the first degree by Mr« Lmbnii»§ who makea Hae of this
notation in hia aaawer to th«t letter. The rectangle contained
by AQ and oa ia eqUai to the rectangle contaided by At' and
aif I and hence mka for determining the fluxions of auch ex«
penentkl <]uantltiea may be dediicfed by the third and tenth pnn
peaitioiia t bnt we fefiurtbeae to the aeoond book«

]Q9* SuppoiethelogarithmAP(;^.41)to increase nnifontJy ;
let j^M be die apace that Would be described by the motion of
p continued unifonnly, in the time P desBcribea P6 ; and let op,
pm^mn^ nr, n, Src. be a series of terms in continued proportion.
Then, if the ftudcn of the logarithm AP be repreaented by
P6^ the firit» second, third, and all the higher flaxiooa of cjp^
dMtt bei^qpteacKited letpeAifdy by;Mbfm, »r, and the ^ubae*

qbent

1

170 Of Logarithms, and the ITusions 'Book'L

qneiit lams bf that geometrical progression. For, since jwiis
to op in the invariable ratio of P6 to oa, it follows^, from the
fifth general theorem^ that the flaxion ofpmh measored by a
line which is to pm (that measures the fluxion of op) as PG is to
OAjtOras jmisto op: butfniiis^chaline^becaQse.mnistojMit
as pm is to op i and therefore mn measures the fluxion of pm, or
the second fiuxicm of op. In tl^e same manner it appears, that
nr measures the fluxion of mn, or the third fluxion of op ; and
that each higher fluxion is measured by a oorresponding teoa
of the progression* It is evidentj thp^t when op is detensin^,
and pm (which ineasures the first fluxion of op) varies^ then
mn (which measures the second fluxion of op) varies in the dur
plicate rat^Q oipm; nr varies in the triplicate ratio oipm ; and
each term of the progression varies in the same proportion as
the power of pwi that has its exponent equal to the number of
terms which precede that term in the series.
• 170! We have sho\vn above, how to distinguish the incre-
ment of the fluent into such parts as may be conceived to be ge-
nerated in consequence of its rcspectiye fluxions, and that bear
a constant relation to their measures, in those cases when the
fluent has no higher than first, second, and third fluxions. In
order to iUustrate what relates to the higher orders of fluxions
Xwhich is rq)resented as v^ry abstrii^)^ we shall now chuse an
instance where fluxiofts of any order take place, and we never
come to an invariable fluxion. Let oq, oa, and op (JigAi) be in
continued proportion; and, op being greater than oa (which
we suppose invariable), let the motion of ^ be unifmm : then
the velocity ofp shall increase in the duplicate ratio of op, be-
cause it is to the velocity of ^ as op is to oq (wet. l67X or in the
duplicate ratio of op to oa. While q describes hi, let j9 describe
ck ; and, oh being tooa as oa is to oc, and o/ to oa as oa is to
oky it follows that ok is to oc as oh is to o/, and ck to oc as A/ is
to oL Let cd he to dk, de to tk,ef\jofk,fg to gk, as A/ is to
ol; and the lines cd, de, cffg, gx, shall be in a geonietrical
progression that may be -continued at pleasure. The sum of
the terms of this progression approaches continually to the
increment ck as the series is produced, so that their diflerence
may become less than any assignable line. The liQe ok being

sup-

supposed ' to flow imifomdy> let its fiuxioa be represented by
hl\ and the firat^^oond, third, fourth^ and all the higher fiux-
ions of oe>shaU be represented respectitely hy ed^^dcj 6ef,24fg,
tod the subsequent terms of theprogressioa having ooefficieuts
prefixed to thm. that arise by the continual multiplication of
the numbers 1, % 3, 4> 5, 6, 4*c. The line ck which is the ior
crementof theflu&ntoc consists of the parts cd, de, ef,fg,gx,^.
and we approach continually to its value by adding to cd, which
kepnsenta the first fluxion of qc, one half of the line that repre*
jenfts ita second fluxio«)^> one sixth part of the line that rcpre-
aents its tbild fluxion, one twenty-fourth part of the iine that
fcpresants its fourtli fluxion, and so pn. The first part cd is
whut would hnvebeen generated, in the timep and q describe
4:^: and A/, if the motion of p had been continued uniformly
from the term when it comes to c,* the sum of the two first
parts cd and de (viz. cc) is what would have been generated
in the same time, if the acceleration of the motion of p had
been continued uniformly from the same term, as in the 74th
and 97th articles ; the sum of the first three parts cd, de, and ef
(jdz* cf) is what would have been generated in the some time
if the increase of that acceleration had been continued uniform*
ly, as in the. 130th and 134th articles, or the line that repre-
sents the second fluxion of op had increased uniformly ; and so
on. To demonstrate these things, suppose oa^ oc, om, on, or,Spc.
to be in continued proportion ; then cd shall be to om as JU is
to oa, dt to 0J» in the duplicate ratio of hi to oa, efto or in
the triplicate ratio of hi to oa, and any term of the series cd,
de, ef,fg, gx, Src. is to the corresponding term of the series om,
an, or, o$, ^c. in a constant ratio, because hi and oa are invari^
able. Theref(»re, by the fifth general theorem, the fluxion of
cd is to the fluxion of om as cd is to piti ; and, in general, the
fluxions of the corresponding terms in those two progressions
are in the same ratio as the terms themselves. But^ by art. l6d>,
the fluxion of om is to the fluxion of oc as 2om is to oc, the flux-
ion of a» is to the fluxion of oc as Son is to oc, the fluxion of
x»r is to the fluxion of oc as 4or is to. oc ; and so on. There-
fojre the fluxion of cd is to the fluxion of oc (which is repre-
sented by cd) as 2cd is to oc ; and, 2de being to cd in the same

ratia

n

ire Of L(pgaf4ikm, m^ tkB JRuwionM ' Bpokl.

r»tio of Sci to 0t, it foUcfvn, Aki Hm fluxioft ^ed i% iqm*>
tented by 2<&. The flaxi^ti of ^e is to the fluxioti of oi; as ^
18 to oc^ and thertfote b^repreiMtiited by Sef; and tfaas it. ap^
peaK; that tbe flexions of oc, cdy de, (f,Jg> ^ are rqnrMentcd
respectively by ed^ 2de, Sef, 4/^> 5gx, i^e. Thegecand iaxiM
of o6 is the dame as the firtt fluxion oiod, and is tfaeveiDre ia^
presented by dde. The tliird fluxion of oc is tha same as the
first fluxion of Q^s ^^^ therefore is represented by 6^« Vbe
fourth fluxion of oc is the first fluxion of &/, and is raptesMHii
by Ufg ; and so on. Whik 6q decreasesap incraasea ; mdi,
the fluxion of oq being eon^d^d M negative^ all the iax«>
ions of op are positi^. When oq increases^ qp dacnMsat)
and^ when j9 comes to e, all the fluxions of ix; are representad by
tixe same quantities as before ; but they are ahernately fiog«*
tire and positiTe when the fluxion of 09 is intariable.

PROP. XII.

171. Let op be greater than oa, ad to ap as oaU to op
(fig. 43) ,• and kt oa, ad, de, e/,/g, Sec. he in continued pro-
portion : then, by adding together ad, idt, \ef, ifg, &c.
tre approximate continually to the ualue ofAP the logarithm
of op.

iLet op, oa, and oj be in eontinued prapoftion, and the motion
of 7 from m towBrds a be uniform^ as in the last article* Then^
be^se aq is to iij> asaa i*to op^tfd is equal to «fy and the ve*
locity of rf is equal to the consUnt tneloc^af f. The veioei-
ty ef P is to the velocity of p as oa is to op (by art. iSl)^ aad
tile Telocity of j» is to the velocity of ^ (aril) as op is to #9
(by art. 197). Therefove the velocity of P is to the velocity
of if as aa is 09^ or as ap is to oa ; so that the motioti of P
increases in the same ratio as ap increases* .Tbefluxioaof AP
is to the fluxion of ad in the sameratio of op to a« ,* aad the
terms oa, ad, de, ef,fg, i^t. being in a oontimted geometrical
progressionj the first term of which oa is invariable, it folkma^
from the eighth proposition^ tbu theflaxiooof «f if tofiiaiiw*

ions

Gbttp^VI. if logarithmc Quamtkiei. 17s

iooiof ye, ^ef, ifgi i^e. ropectifvcb^, MMutoad^de, tf,
4^. Tktfcfon theflosicn of AP i» to the flaxioM of ad, ide^
fO^ i^» ^ n^ectmly, as op is to the t«noB om, ^d, de, c^
^, BfA, by adding togetliftr the terms oa, ^mI^ ife^ ^ ^. wt
affM^dmalc ooaitiiiuaBy to op ; ibrml is lx>ii^, det0€p,4fto
fy, ^c. a« Mift to op; and the differenoe between op and the
son of the lame oa, md, de, ef, ^c. mssy become lets than aof
•iiagnRble magaitiide^ by eoatimung the progresskm. Tbei«»
fore^ by adding leather the flaxions of ad, i4c, \ef, \fg, ^-e.
we appoxunate oonti&ttaUy to the Tahie of the flaxion of AP;
aad^ oonaBqaently^ by summing up the lines ad, ^t, ^, \fg,
4rc. we approach oontinuaUy to the Talae of AP the logarithm
rfop.

PROP. xni.

172. Tke same tldngs being mpposedasin the last proposition,
we approximate contimtally to the logaritkm ofod hysum^
mmg up the dijf^erencts betwixt ad and frfc^ \ef, and \fgj
\gh, and \h3:, -&c.

Let od, oa, and 02: be in continued proportion j and oq shall be
to o;2;aso£/isto(>p; sothat};rshallbetodfpaso<^rsto od^. Let
jz be divided in the same proportion in the points m, n, r,f,S^c.
a$ dp is divided in the points e^y) g, h, Spc. and the sum of the
terms oq, qm,mn, nr, r/^t^cshall approximate continually to oz.
But as the difference of oa and ad (ox aq) is equal to oq, so the
difference of de and ef'is equal to qn : for de is to efas oa is to
aq (or ad), by the supposition ; and, consequently, rf^is to the
difference betwixt de and efas od is to oq, or as c^ is to qz, and,
therefore, as df is to qn. In the same manner, the difference of
fg and gh is equal to n/"; and, in general, the difference of any
two terms which immediately succeed each other in the pro-
gression de, ef,fg, gh, Sfc. is equal to the sum of the two cor-
responding terms of the progression qm, mn, nr, rf, S^c, There-
fore, by summing up the differences betwixt oq. and ad, de and
^fifg ^^^ gf^y ^'^' w^ approximate continually to the value of
oz. But the fluxion of AD, the logarithm of od, is to the

fluxion

n

174 Of Logarithm, and the Ftuxions Book t^

fluxion of od (or o£ad)sBoa istood (by ait. ldl)> or aswis
to oa ; and the fluxion of ad (by prop. 8) is to the fluxions.of
the terms ad, {de, {ef, \fg, ^h, ^ik, ^c. respectively, as oa
is to the terms oa, ad, de, ef, fg, gh, Sfc* Therefore we conti*
anally af^roximate to the value of the fluxion of AD by snm*
sung up the fluxions of the diflerences betwixt od and jde^ ^f
and \fg, \gh and \hk, ^c. and consequently^ we appioximale
to the value of AD, the logarithm of od, by summing tip these,
diflerences themselves.

173. Cor. Let oa be to oei as op is to ox ; and the logarithm
of ox shall be equal to the sum of the logarithms of op and od;,
to which, therefore, we approximate by summing up Qad,^f^
^h, Src. It is manifest, that ox is to oa as od is to o; ; and that
the logarithm of ox is that which measures the ratio of od to
oq. But od and oq have half theii^sum equal to oa and half
tlieir difierence equal to ad, which are the two first terms of
the geometrical progression oa, ad, dt, tf,fg,gh, S^c. and it is
obvious, that what has ^n shown of these logarithms coin-*
cides with whattlie excellent Dr. HaUey demonstrates in the
Philosophical Transactions, num. 216. This would lead us to
the method of computing logarithms, but we refer that to the
next book.

174. Hitherto we have supposed with Lord Napier, in his
first scheme of logarithms, that, while op increases or decreases
proportionally, the uniform motion of the point P, by wlxich
the logarithm of op (or the measure of the ratio of op to oa) is
generated, is equal to the velocity ofp at a, that is, at the term *
of time when the logarithms begin to be generated. But the
uniform motion of P may be supposed equal to the motion of
p at any other term, as when it comes to c ; in which case the
constant velocity of P is to the velocity witli which p sets out
from a when the logarithms begin to be generated, as oe is to
oa ; and thus we may have as many systems of logarithms as
we please. Tlic properties mentioned in the 156th and 157th
articles, by which they become useful for facilitating computa-
tions, are common to all the systems. Tlie Bne oe is what the
learned Mr. Cotes calls the Modulus ofiiie system. The mea«
spres of a given ratio in the diflerent systems arc in the same

. proportion

Chap. VI. of logarithmic Qu^mtUut. t7A

proportion as the lines ot, t being ahrays die tain irbei» the
velodtj oSp becomes equal to the constant Telocitj of P« The
logarithm of any quantity in Napier^s first system become^
equal to the logaritlnn of the same quantity inauyother system^
whose modulus is oe, by mnltipljring it by the nuidber which
erpiessestheratioof oe to oa; and the modulus of any system
is tothemodufaisof any other system, as the logaridun of any
given quantity in the first is to its logarithm in the second*
Thus, in Napier^s first scheme, in the same time that up firom
being equal to oa becomes equal to ten times oa, the point P
describes a line the ratio of which to da is that of 2.3005809^
ifc. to unit^ But it was found convenient, that the logarithm
often should be supposed equal to .unit ; and the motion of P
was supposed to be so far diminished, that the space described
by it in that tune mightbe equal only to oa ; that is, its veloci*
ty in this case was supposed less thatf its vdocity in the former,
in the same ratio as I is less than d.d025609, 4^^. If or be taken
less than oa in the same ratio, the velocity of P shaU be
equal in this case to the velocity ofp at e ; and oe shall be the
modulus of this system, vrfiich theiefore is expressed by
0.434£g44, Src. oa being unit.

P

F

1 75. When a ratio is given, its measure is always as the nuK
didus of the system* It is therefore the same invariable ratio
that is ahrays measured by the modulus of the system, which
is by Mr. Coies called the ratio tnodularis. This ratio is that
of oiffi to Of, if op by increasing proporUonally from being equal
to oc become equal tq om, in the same time that P by its uni*
form, motion describes a line equal to the modulus: oe*

176. WliiletbebaseOP(j^g.44)mcreasesuniformly,lettheor*
dinatePpiacreaseor decreaseproportionally (that i6,ktthevelo«
city of p iid the direction Oo be always as the i^dinate Pjp) ;

and

1

ITS O/LogtuiikmsymiihiBuM Book I*

Mi fkBfomlp nhJl deaoAe die bgoriihmic curve* The bait
QP is flfarqfA theiogsfitiwi of the otMwtitt fp ; aad^ if Ae

logarithsiB ; indthefloxkynofflOf •rriiuleislotJbeAinmcf
dieb«aeafl.tliatordi3iateb toOo. llieoidiMtaislc^pttldifl^
iBDccs fiiam eac^ etfasr an ^ir&p in gTMcisfcdad pn^^
fiaoBiiriiidi it fioDiNra^ diat wlMi thekaaeaiid omw irefpiDr
jhwrrl, riifiy nppmnrh tn nafth nriwr TTn»*fmr>lj rm rut liiir^ nn
tbaft tiieirdhlMioe aMy btmtie Ins tluor amy aaBigiiaUe Mne ;
Imtbaeaofla tbey can never maety thdiaae it thttiefiMre anaspnp^
toleof the curve. . When mof tma poinia df die enrae, aa e
ani^ aKgiwen^yonnvif deloranneeaaaanymoiiepoai^
fllieme,hymm9igcfo6SmaMe»vfmkA^

aminamy gaoaietrioal prdgseMio^^f vbdehtiieaffdixiataBilajaad
ikrf ase any aire teima. Whantiie carve it sapfMiaad to W d^
tcrflbeiiy die axpeaentiai qtianaif ipb aaa easi^ deberauoed by iL
LettheoTdiaateAebaeaptetatdbyA; letOPbeioOAasany
ipiancityespietasdbyjistoumit: and the ordinate Pp may ke
MJpiatttd jy A^, att enpanential quatrtity of die firtt degiee
when X is variable.

177 • Let Imn (Jig. 45) be an hyperbola^ o its center^ or and or its
asT^pto tes ; let aljpmj and ^ parallel to the asymptote or meet
the curve in I, m, and $L Let AL be equal to al, the angle LAP
equal to lap, and AP be always the logarithm of op according
to the 153d and I54th articles j then the hyperbolic area almp
thall be always equal to the parallelogram LP. For^ if pg be
to P6 as the velocity of p is to the velocity of P, that is (by
art V6l), as op is to otf^ er as «/ is to jpm (^ the property of
Aehjrperb^; -dien idiaHp^ betoFGr aaPM ieta fm^fihr
.psrrilekMgrsin mg dheil be abrafs equd f» the partielogfaia
MG, andfbypiop. 4Odie4aKiin0f tfaebypeiATOlicafea^lMjp
equal lo d^ fiction «f the paraSeAogrMi LP. TheKfoie, by
the fenifh general theoran^ tlie byperb<dtc aiea citip ia equrf
to the paraMelegram LP^ and the area cAnp^ or the teetar oaif
<irt^eh is eqn^ to it^ beeaone the trijinglesepiTi^ •««/ are equal),
a e a sates die rsitio af op to eg, Tbeae aveas ferm a ay Mem of
logaridnns^ tkemodulos of which is Ike p aaa H eiegfam el, and

serve

Chap. VL ^flogntiihmie QaakHHei* 177

mxwe for measaring rattoe^ as the arches or sectors of circles
aerve (ox measaring angles. The former are divided into equal
fMurte, by resolving a ratio into the equal ratios from which it is
Aompouaded ; as^ by dividing the latter into equkl parts^ an
angle is resdved into the equal angles of ^hich it consists.

1 78* The area of the parabola admits of a perfect quadrature^
a&d elliptic aieas are reduced to such as are circular ; therefore
the areas of all the conic sections may he reduced to rectilineal
£gures^ the measures of angles^ or the measure^ of ratios; and
to one or mote of these all fluents ought to be reduced in this
doctrineas much as possible. As a conic section passes from one
species to another, by varjang the inclination of the plane that
cuts the cone ; or by varying the circumstances of the descrip*
tion^ when the curve is traced upon a plane by motion ; or by
changing a sign^ coefficient or exponent dl a term, when the
matare of the curve is expressed by an equation : so the expres-
sions of the measures of angles and of ratios are by similar vari-
ations transformed into each other, and in some cases into such
as represent rectilineal figures ; and, by a change in. the sigUj
coefficient or exponent of a term in the expression of a fluxion^
the nature of the fluent is so far altered that it becomes reducible
to a conic section of a difierent kind. When cp increases or
decreases proportionally, the aoceleration or retardation of the
motion of j9 is as its distance from the given point o ; but there
are also other cases in which its acceleration or retardation ob-
serves the same law : and, by taking these together, we may
comprehend the genesis of the lines that measure ratios and
«iiigks in one ^ew. But we shall have occasion afterwards to
consider ftirther the analogy there is betwixt those measures,
end to treat of the logarithms that are called imfiginarrf.

179. In Napier's first system, the constant velociy of P (fig, 46),
by which the logarithm AP is generated, is equal to the velo-
ciiy with which p sets out from a when their genesb is suppos-
ed to begin (art. 153 and 154). Therefore, if we suppose AP
and Off to be diminished continually (AP being always the lo-
garithm of op), their ratio shall approach continually to a ratio
of equality as its limit (by art. 6?) ; and, aa being supposed to
represent unit, if ap be very small compared with oa^it may be

VOL. I. M supposed

1

178 Of the Tangents of curve Lines. Book L

supposed in approximations to bethelogarithm of op in this sys-
tem^ which is called its hyperbolic logarithm; and any very small
fraction may he supposed to be the logarithm of the sum of
' unit and that fraction added together. From this it follows^
that^ if a series of mean proportionals be interposed betwixt oa
and any given line ob, and the number of all the terms without
including oa be x, and on being the second term of the series an
be to aft as unit is to the number q ; then, by increasing conti-
nually the number of mean proportionals betwixt oa and oby the
ratio of x to q shall approach continually to the ratio of AB
(the logarithm of ob) to aby as its limit* For the number x is
to unit as AB is to AN (the logarithm of on) ; and unit is to q
as an (which approaches continually to AN) is to ab. There-
fore the ratio of x to 9 approaches continually to that of AB to
ab. For example ; if 06 be double of oa, the ratio of j: to q*
approaches continually to the ratio of the logarithm of 2 to
unit^ which is nearly that of 7 to 10.

CHAP. VII.
Of the Tangents of curve Lines.

180. An arch of a curve has its concavity turned one. way,
when the right lines that join any two of its points aie all on
the same side of the arch ; or in general (that we may include,
with Jrckimedes, f such lines as have rectilineal parts), a line
has its concavity turned one way, when the right lines that
join any two of its {)oints are either all upon one side of it, or
while some fall upon the line itself, none fall on the opposite
side.

• Thw, if the fnction 1 + J (repreiented here Vy on) raised to the power
jrbe supposed equal to 2, the ratio of x to ^ thall be nearly that of the lo^rithm of 9
to unit, when j ia a laTg^e number. See the J)actrittg of Chances^ probU 5, by the
excellent Analyst Mr. de Moiyre.

t De iphcn dr cylindro, defin. 2.

181. As

PJa1e>I Pa,/7SyalJ.

*

Chap. VII. Of the Tangents of curve Lines. 179

181. As a right tiae is the tangent of a circle^ when it touches
the circle so closely that no right line can be drawn through
the point of contact betwixt it and ihfi arch^ or within the
angle of contact that is formed by them ; so, in general, when
a right iline ET (Jig. 47) touches any arch of a curve, as EH in
£, in such amanner that no right line can be drawn through E
betwixt the right line ET and the arch EH, or within the
angle of contact HET that is formed by them, then is ET the
Tangent of the curve at E. It is manifest, that the right lines
ET and EH are on different sides of the arch EH ; and that
when the arch has its concavity turned one way, the tangent
at any point of it is on the convex side.

182. The rightlineT£(^.48)being continued toe,ifEf is the
tangent of the arch EC the continuation of HE, then the arch
HEC has a continued curvature at E. When the arches EH
and EC are on different sides of the tangent TEt, the point E
is called a point of contrary flexion. But if any right line ER
(fig* ^9)f different from Et the continuation of ET, touch the
arch EC, then the point E is a double poiftt of the curve, and is
the intersection of two arches which have different tangents at
that point, or are on opposite sides of the same tangent, and in
some cases on the same side of it.

183. When two lines that have their concavity turned the
same way have the same terms, and the one includes the other,
or has its concavity towards it, the perimeter of that which in*
dudes is greater than the paimeter of that which is included.
This is the second axiom of the treatise of Archimedes con-
cerning the sphere apd cylinder.

LEMMA VH.

184. The base being suppoud to flow uniformly, the ordinate
increases with a motion that is continually accelerated, and
decreases with a motion that is continually retarded, when
the arch is convex towards the base. But when the arch is
concave towards the base, the ordinate increases with a rc-
tarded motion, and decreases with an accelerated motion.

Mst U,

\

200 Qjf the T^*ni$ f^ curve Umi. BookL

Let Ibf ig^^h be convex towwrdt Ae base, and kt ilie ordinate
Tp (fig* 47) increase while the base APincieaaea. Let BD aad
DG be equa) lines described by P with an anifonm motion in asj
equal tiniestlMtimQiediateiyflaeoeed after each other. Let the
ordinates BQ, JXB>, andGH meet the curve in the pointsC^Bt
liid-H ; let t^ tangent at £ meet the ordinatesGH and BC
in T and / i and let M£I parallel to the base meet these or*
dinates in I and M* Th«, because BD and DO are equal, IT
U e^ual to Ml I and»the arch C£H being convex towards the
baas, IH is greater than IT, or M^ ; which is itself greater than
DdC, for the same reaflon. Therefore IH is surely greater than
MC. But MC and IH are the incranealsof the ordinate thi^
are generated m eqnal times while the base acquires the aug-
ntentsBD and DG ; aud^ since those increments continually in-
iftrease, it follows, thai th^ motion with which the ordinate flows
ia continuaUy accelerated. When the ordinate decreases, the
decremfsntsIH^MCcontinuaUy decEease,aBul the oniinaie flows
with a motion that is eoAltniwdly retarded*

185. I^ the afch be concave lowaids the base ; and, the
construction being tba samie, MC (fig. 50) shidl exceed Mf , or
IT> which is itself j^alor Aan IH: so that MC being gieater
than IH, the ordinate flows with a motion that is retarded or
accelerated continually, a^ccording as it increases or'decieases
while the base increases.

186. Themotion of thepointjpOEg.47),which is snpposedto
decribe the curve, is peipetaaUy accelerated, when, the base be-
ing supposed to flow uni£omdiy, the curve is convex towards
the base, and the ordinates increase whik the curve increases.
For the arch EH being greater than its chord EH, which in
this case exceeds the tengent ET (because the point T is be-
twixt I and H, and the angle ETH is greater than EHT), the
arch EH b surely greater than the tangent ET. But the arch
EC is less than the tangent Et, or ET ; for if CV panHel to
the base meet the tangent Et in V, VC shall be less than V^ ;
and, the arch EC bemg less than lj[ie sum of EV and VC (by
art. 183), it nuist be less than Bt. Therefore the arch EH is
greater than CE, and the motion of p in describing the curve is
perpetually accelerated. In Ibeiuune maanei: it appears, that,

when

Chmp. Vn. Of the Tiange9M of curve Llna. )81

wfaea the cunre it cODTex towaids the base^ and the onliiiatei
decrease while the cmre increases^ the motion of p(or thai with
which the curve Fjp flows) it retarded perpetuaUj.

187. But when the curve it concave towardt the bate^ andi
the bate being tuppoted to flow uniformly, the curve increaiet
while the ordinatet increase; the arch EC (fig. 50) it greater
than the chord EC, and therefore greater than the tangenlEf,
or ET, which it itself greater than the sum of £V and VH
(VH parallel to the bate bdng supposed to meet the tangent
in V), and consequently is greater than the aroh EH (by art
1 83). Therefore, in this case, the lines described bjp in equal
succeeding times perpetually decrease, and its motion (or the
motion with which the curve Vp flows) is peipetually letard*
ed. But if the ordinates decrease, in this case, while the curve
increases, the motion of p is perpetually accelerated.

PROP. XIV.

188. Let ET'be the tangent of the curve VEatE; and, EI
being paraUtl to the ba»e AD, let IT bepardUel to the or^
dkkite DE : then, the ftuxiom if the base, ordinate, and
curve, ehall be meamred by the right Nme EI, TF, and £T>
retpectivefy.

First, let the arch CEH C/^. 47) be convex towards the bate ;
and, the base beingsuppo8edtoflowuniformly,theordinateshall
increase with an accelerated motion, by the lastkmm^ The
figure and construction being the same as in the 184th article^
it follows, firom the first and second axioms, that, in the same
time that the base acquires the augment D6, a line less than
IH,but greater than MC, would be generated by themotionwith
which the ordinate DE flows continued unifonnly* There*
tare, if Uiis line be not equal to IT, first let it be equal to
some line 11^ less than IH, but greater than IT. Join EL, and
it shall meet the curve EH in some point R betwixt £ and H ;
let the ordinate RQ meet the tangent in N, the base in Qj
and theright line £1 in K. Then^ unce XL is to KR as EL

Ms is

18€ Of the Tangents of curve Line$. Book I.

is to £R^ or as DG is to DQ^ and^ when the generating mo*
tion is uniform^ the quantities generated are in the same pro*
portion as the times ; it follows, that, if the motion with which
the ordinate D£ flow? was continued uniformly, the line KR
would be generated by it in the same time that the base ac*
quires the augment DQ. But, because the same line KR is ge*
nerated by the accelerated 'motion with which the ordinate
flows, in the same time that the base by flowing uniformly ac-
quires the augment DQ, it follows, from the first ^xiom, that
a less line than KR would be generated in that time by the
motion with w^iich the ordinate D£ flows continued uniformly.
And these being contradictory, it follows, that the line that
would be generated by the motion with which the ordinate BE*
flows continued uniformly while the base acquires the aug-
ment DG, is not greater than IT. If it be said to be less
than JT, or Mt, let it be equal to M/ ; and, M/ being greater
than MC, but less than M^, it follows that E/ must meet the
arch CE in some point r betwixt E and C. Let the ordinate
rq meet the tangent in n, the base in q, and the right line EM
in k. Then, since Ml is to Ar as EM is to EA, or as DB is to
Dq ; it follows, that, if the motion with which the ordinate DE
flows was continued uniformly, it would generate a line equal
to kr in the time that the base acquires an augment equal to
D^. But, since the increment kr is generated in an equal
time by the accelerated motion with which the ordinate flows
before p comes to £, that is, before the motion with which the
ordinate DE flows is acquired ; it follows, from the second
axiom, that a line greater than kr would be generated in the
same time by the motion with which D£ flows continued uni*
fortnly. And these being also contradictory, it follows, that
IT measures accurately the motion with which the ordinate
DE flows, or its fluxion, when the motion with which the base
flows, or its fluxion, is measured by DG or EI.

169. li^ the same case the motion of the point jti, that is sup-
posed to describe the curve, is perpetually accelerated, by art,
}86. If the motion of p was continued uniformly from E, a
line Jess than the arch EH, but greater than the arch CE^ would
be described by it in the time P describes DG, by the first and

second

Cbap. VII. Of the Tangents of curve Lints. 183

second axioms. If this line be not equal to the tangent ET^
first let it exceed £T by TL ; and^ because this line is less
than the arch EH^ which is less than the sum of ET and TH
(by art. 185% it follows, that TL is less than TH, and that
the right line EL must meet the arch EH in some point
R betwixt E and H. Let the ordinate RQ meet the base
in Q and the tangent in N. Then, since ET is to EN, and
TLtoNR, as DGistoDQ; it follows, that the sum of ET
and TL is to the sum of EN and NR as DG is to DQ ; and
that a line equal to the sum of EN and NR would be describ-
ed by /7, if its motion was continued uniformly from the term
when it comes to E, in the time P describes DQ. But, while
the motion of P is uniform^ the motion ofp, in describing the
arch £R, is perpetually accelerated (by art. 186), and it fol-
lows, from the first axiom, that, in the time P describes DQ,
a line les^ than the arch ER (and consequently less than the
sum of EN and NR) would be described by p if its motion
was continued uniformly from £. And these being contra-
dictory, it follows, that the line which would be described by jr
if its motion was continued uniformly from E, in the time P de-
scribes DG, is not greater than ET. If this line be smd to be
less than ET, or E^, let E^ exceed it by tl ; and, since that line
is greater than the arch EC (which is greater than the chord
' EC, and therefore is greater than the excess of £^ above ^C), it
follows, that tl must be less than f C, and that the right line E/
must meet the arch EC in some point r between E and C. Let
the ordinate rq meet the base in q, and the tangent in n : and,
since E^ is to En, and tl is to nr, as DB is to D^ ; it follows,
that, if the motion of p was continued uniformly from E, a
line equal to the difference of En and nr would be described
by it in the time P describes a line equal to ^D. But, the mo-
tion o(p being perpetually accelerated, the same Une must be
greater than the arch rE(by ax. 2), and, therefore, greater than
the difference of En and nr, which is less than Er, by art. 183.
And these being also contradictory, it appears, that the motion
of p at the term when it comes to E, or the fluxion of the
curve FE, is measured accurately by ET when the fluxion of
the base is measured by DG, or EI.

M 4 190. Wheii

n

184 Of the TangetUz of eiiroe Um^ Book I«

190 (Fig. 50). When the curve is concave towards the baie»
the proposition is demonstrated in the same manner from the
third and fourth axioms; oritmaybe deduced fifom the precede*
ing case^ by drawing anyrightlineaoparalleltothebaseinsttcb a
manner that the arch C£H may have its convexity towards it,
For^ if D£ produced meetoo in d, and the curve meet no in/^ the
motions with which <2E and/£ decrease are equal to those with
which D£ and F£ increase. When the base flows with a vari^r
able motion^ the proposition is demonstrated from what we bavo
shown by the eleventh general theorem^ and the 60th article.

191. Cor. (J^^,47) In the first case> when the curve is convex
towards the base, the partTH of the increment IH of the ordi-
nate D£ that is generated while the base acquires the augment
DO, is owing to the acceleration of the motion with which the
ordinate flows during that time. When the curve is concave to*
wards the base, the increment IH0{g.50)of the ordinate is lest
than IT ; and the difference arises from the retardation of the
motion with which the ordinate flows. The fluxion of the oidi**
Date is the same in all curves that have the same tangent at E,

192. The point S (fig. 6\\ the right line AO, and the circle
FBBdescribedabout the centres, being givenin ]^osition,ktthe
point P describe the right line AO, SP meet the circle FDB in
N; and let the point M describe any right line SD so that SM
may be always equal to SP. Let SA be perpendicular on AO (
and^ if the motion of the point P from A towards O be uni-
form, the motion of the point M in the right line SD shall be
perpetually accelerated ; but the motion of the point N shall be
perpetually retarded. For let A£, £H be equal lines described
by P in any equal times. Let SA, S£, and SH meet the circle
FDB in g, D, and 6* From S as centre, through k and H, de<«
scribe the arches Kk, HK meeting S£ in k and K, and let hi,
HL be perpendicular on S£ in / and L. Then shall AE and
EK be the lines described by M, and gD, D6 the arches de-
scribed by N in the same equal times. Because AE is equal to
EH, therefore (Elem. fi6. 1.) £/is equal to EL; and, EK
being greater than EL, or £/, which is greater than Ei, it is
evident that EK is greater than EA; and, therefore, the motion
of M is perpetually accelerated. Because HL is equal toA/^bot

SH

Chap. Vn. Cftie TangenH ofcune Una. 185

SH greater dian S&, theangkHSLis lev than AS/^ and tbe
motion of N is perpetually retarded. It is obvious^ that^ whett
the motion of M iram S towards N is uniform^ the motions of
P and N are retarded ; and that, when the motion of N ftom
/"towards b is uni^Dnn, the motions of P and M are perpetually
«cceler«led« '

PROP. XV.

193. The paha S (fig. 52) and the right line AO
given in position, let the circle fUb described from the centre
S meet AO in E; and the fluxions of the right Knes A£, S^
and of the archfE shall be to each other in the same ratio
as the right lines SE,AE, and SA.

While the point P describes AO, let SP meet the circle fEb
in N, and upon SB let SM be always equal to SP, as in the
preceding article. Then the fluxions of the right lines AE^
S£, and of the arch/E shall be measured by the velocities of
the points P, M^ and N, at the term when they come together
to E. ' Let the motion of P be uniform, and the motion of M
shall be accelerated by the last article. It is manifest, that the
Telocity of P is greater than the velocity of M. If the vdocity
of P be to the velocity of M, at the term when they come to
£, in a less ratio than that of SE to AE, let their ratio be tba
same as that of SH to AH, or (EK being perpendicular on
SH in K) that of EH to KH ; and the point M shall describe a
line equal to KH, if its motion be continued uniformly from E^
in thetimePdescribesEH; and,KHbeinggreaterthanGHthe
excess of SH above SE, it follows, that a greater line shall be
described in the same time by M, when its motion is continued
uniformly from E, than when it is continually accelerated from
the same term, against the first axiom. If the velocity of P be
to the velocity of M at the term when they come to E in a
greater ratio than that of SE to A£, let their ratio be the same
as that of SC to AQ or (£A bemg perpendicular on SCg in *)

that

1 Bfi . Oftke Tangents of curve Lines. Book T.

that of EC to Ck; and the point M shall describe a line equal
to Ck by its motion continued uniformly from E in the same
time P describes a line equal to C£ ; and, CJk being less than
Cg the excess of S£ above SC^it follows, that the line describ-
ed by an accelerated motion is greater than the hue which is
described in an equal time by the motion acquired by this acce-
leration continued uniformly, against the second axiom. There-
fore the velocity of P is to the velocity of M, at the term
when they come to E, as SE is to AEl This might have been
demonstrated from the g6th article, in a different manner,
because the fluxion of the square of SE is equal to the fluxion
of the square of AE.

194. The ihotion of P from A towards O being uniform, the
motion of N is perpetually retarded, by the ig2d article. If
the velocity of P be' to the velocity of N as SH greater than
SE is to SA, or as EH is to £K, the point N shall describe
£K, by its motion continued uniformly from £ in the time P
describes EH ; and, £K being less than the arch EG which is
described in the same time by N with a retarded motion, it fol-
lows, that a less line would be described in the same time by
a motion continued uniformly, than by the same motion perpe-
tually retarded from the same term, against the third axiom.
If the velocity of P be to the velocity of N in a less ratio than
that of SE to SA; or (CR being perpendicular in R on ET the
tangent of the circle at E) that of CE to ER,let it be the same
as that of CEj to Ex; join Cx, and let Sz parallel to it meetAE
in Q, the arch Eg in ti, and the tangent ET in z. Then, since
Ex is to £z as EC is to EQ, it follows, that the point N would
describe a line equal to Ez, by its motion continued uniformly
from E, in the time P describes a line equal to EQ. But the
point N describes the arch wE by its retarded motion in that
time before it comes to E ; and the arch uE being less than its
tangent Ez, it follows, that the line described by N while its
motion is retarded, is less than the line which would be describ-
ed in an equal time by the motion that remains after that re-
tardation continued uniformly, against the fourth axiom. There-
fore the velocities of the points P, M, and N, at the term when
they come together to £, are in the same ratio as the right

lines

Chap. VII. Of the Tangaits of curve Lines. 187

lines SE^ AE, and S A ; and tlie fluxions of the right lines AE^
SE^ and of the arcb/E are in the same ratio.- This may be
shown in like manner, by supposing the motion of M or'that
of N to be uniform ; and the proposition is made general by
the eleventh theorem.

195. Cor. I. Let a circle A«l) described from the centre S
meet SP in n, and SE in D ; and the velocity of n shall be al-
ways to the velocity of N as Sn or SA is to SN or SE,'by
theor. 3. The velocity of N is to the velocity of P whea
they come to E in the same ratio. Therefore the velocity of
n at D is to the velocity of P at E in the duplicate ratio of SA
to SE ; and the fluxion of the arch AD is to the fluxion of its
tangent AE in the same ratio. In the same manner it appears,
that the fluxion of the arch AD is to the fluxion of its secant
SE as the square of SA is to the rectangle SEA.

196. Cor. II. If the points P and p set out with equal velo*
cities from A and a in the right lines AO, ao at the same time ;
and, the motion ofp being continued uniform, the velocity of
P be always as its distance from S ; then ap shall be the loga-
rithm of the sum of SP and AP. For the fluxion of op shall be
to the fluxion of AP as SA is to SP, and to the fluxion of SP as
SA is to AP. Therefore the fluxion of ap shall be to the sum
of the fluxions of SP and AP as SA is to the sum of SP and
AP ; and, when ap increases uniformly, the sum of SP and AP
increases proportionally (by art. I60). Therefore ap is the lo-
garithm of the sum of SP and AP, or is the measure of the ra-
tio of that sum to SA, the modulus being SA. The velocity
of P is to the velocity of j? as the velocity of p is to the velo-
city of n ; and, when P comes to E, the velocity of p is equal
to the velocity of N.

197. Cor, III. Let the right line ao 0?g.53)given in position
meet AO in F,and SP always intersectaoin/?; then the velocity
of P shall be to the velocity ofp as the rectangle SPF is to the
rectangle SpF; and, if Sa be perpendicular on ao, the fluxion
of AP shall be to the fluxion of op in the same ratio. This
may be deduced from the 15th proposition, or immediately
from the axioms, thus : If ao meet SA in some point betwixt
S and A, the motion ofp shall be accelerated when the motion

of

ISS Of the Tat^tnH of curot lAna. Book I«

of P from A towards O is uniibrm. For^ while P describes any
equal lines CE^ £H^ let j? describe cc and th ; let UL parallel
to AF meet SH and SC in L and / ; and it is manifest, that Le
shall be equal to e(, but that eh shall be greater than tc ; and
therefore the motion ofp is accelerated. Let Uc parallel to SE
meet ehmk; and> if the motion of p was to be continued uni-
formly from c, the line ek would be described by it in the same
time P describes £H. For, if it should he said, that it would
describe ergreater thane/:inthattime,let S; parallel toLr meet
EH in Q, rL in x, and eA in j^. Then, since rr is to ej aseL
isto e;r, or as £H is to £Q,it follows, that the point p, by its
motion continued uniformly from c, would describe eq in the
time P describes £Q. But the point j7 describes the same line
tq in the same time when its motion is continually accelerated
from that term ; so that the same space would be described in
the sametimeby p when its motion is continued uniformly from
c, and when it is continually accelerated, against the first axiom*
In like manner it is shown, from the second axiom, tliat the
line which would be described by the motion oip continued
imiformly from e,in the time P describes EH, is not less than eft«
Therefore that line is equal to tk ; and the velocity ofp is to
he velocity of P, when they cometo^ and£, asei is to EH;
and the fluxion of ae is to the fluxion of AE in the same ratio.
But the ratio of ek to EH is compounded of the ratio of eft to
rl^or that of eFto EF, and of the ratio of eL to EH, or that
of & to SE. Therefore the fluxion of a« is to the fluxion of
AE as the rectangle Sf F is to the rectangle SEP. When the
motion ofp is retarded while that of P is uniform, the demon-
stration is deduced in the same manner from the third and fourth
axioms : or the same demonstration may serve, by supposing the
motion of /> in those cases uniform ; for the motion of P shall
then be accelerated^ When the points E and e come to F, eft
is to EH as SA is to Sa ; and the fluxion of aF is to the flux-
ion of AF in that ratio. When ao is parallel to AO, it is
manifest, that the fluxion of ae is to the fluxion of AE as S<
is to SE, which in this case is an invariable ratio ; and is the
same as that of Sa to SA.

198. Cor.

Chap. VIL €fAc TmgeaOs ofeitrvt lAna. 189

198. Cor. IV. The fluxion of SE is to the fluxion of AB
«B A£ is to SE ; and Hie flaxkm of ae is to the fluxion of Se as
Se is to oe. Therefore the fluxiim of SE is to the fluxion of Sc
u the rectangle AEF is to the rectangle ae¥. Let Su constitute
the angle oStf equal to ASF^ and meet aP in u ; then the flux-
ion of SE shall be to the fluxion of S^^ when the points E and
$ come to F^ as the rectangle contained by AF and Sa is to the
rectangle contsdned by oF and SA> or as ou is to aF«

199* ^rcAimeiies demonstrates O^g.54)^ in theCth proportion of
his treatise concerning q>iral lines^ that a right line SP may be
drawn from S meeting AE in P^ and the circle /E in N^ so
that FN may be to the chord EN in any ratb less than that of
AE to SA.' For> if SI parallel to AE meet the chord NE and
tangent ZE produced in V and I^ PN shall be to EN as SN ot
SE is to NV ; and since NV may beeqnal to any right line
greater than EI^ it follows^ that PN may be to EN in any ra-
tio less than that of SEto EI^ or that of AE toSA. It is also
manifest, that as N approadies to E, NV decreases^ and that
the ratio of SE to NV or of PN to EN continually decreases.
In like manner he shows^ in the 7th proposition^ that a right
line Sp may be drawn from S meeting the right line AE produc-
ed beyond Rm p, and the circle in n, so that pn may be to
the ch(^ En in any ratio greater than that of AE to SA^ or
tliat of SE to El ; because^ if En produced meet SI in v,pn
shall be to En as Sn or SEis tom^^andnvmaybeequal to any
line less than EL In this case^ when n approaches to E, 919 in-
creases^ and the ratio of pn to En or of SE to nv continually
decreases.

200(Ftg.55).Inthe8thpropositionofthesametreatiseheshowB,
that PN may be to the tangent EZ in any ratio less thfn that of
AE to SA> as that of SE to EL, EL being great^ than EL For
let a circle described through L, S, and I meet SE produced in R4
then^ because EL is greater than EI, a right line SZY may be
drawn from S meeting diis circle in Y^ so that ZY may be
equal to ER. Supposing that SZ is such aright line^ the rectangle
SZY being equal to LZI^ and the rectangle contained by SZ
and EI being equal to the rectan^ contained by SP and ZI^
because SZ is to SP as ZI is to EI, it follows, that the rec«
tangle LZI is to the rectangle contained by SP and ZI as the

rectangle

IVO Of the Tangents of curve Lines. Book I.

rectangle SZY is to the rectaogle cootaiDcd by SZ and EL
Therefore LZ is to SP as ZY or £R is to £1, or as EL is to
£8 or NS; and EZ is to PN as EL is to SE. It is easy to see,
that when N approaches to E, the ratio of PN to the tangent
EZ increases continually. In like manner he shows, in the 9th
proposition, that pit may be to the tangent Ez in any ratio
that is greater than that of SE to EI, or of AE to SA ; and
it 18 easy to show, that, when n approaches to E, the ratio of
jpii to Ez decreases continually.

201. Let P describe an arch FPH (/?g. 56) of a continued curva-
ture; let SP meet the circleyiB6 in N; and suppose that SM is
always taken equal to SP, as in art. 19£. Then, if SP increase
contmuaUy, and, the arch being supposed to have its convexity
towards S, if the motion of N be uniform, the motion of jNf
shall be perpetually accelerated. For, if the arch EN be equal
to En, and the right lines SN, Sn meet the curve in P and/?,
and its tangent at E in T and t, Ei shall be greater than ET,.
and the excess of Sp above SE being greater than the excess of
St above SE, which is greater than the excess of SE above ST,
it must be greater than the excess of SE above SP. In this case,
the motion of P in the curve is also accelerated ; for it is mani-
fest, th^t the arch Ep is greater than the arch EP. When the
arch FPH is concave towards S, and the motion of N is uni-
fbrm,the motion of M may be uniform, accelerated, or retarded.

PROP. XVI.

202. Lei tJie circle fEb (fig. 57) meet the curve FPH in E, and
SD perpendicular on SE meet the tangent ET in D ; then the
fiuxioj^ of the. ray SE shall be to the fluxion of the arch fE
m S&is to SD.

Case 1 . The motion of N in the arch /Eft being uniform, let
the motion of M in the line SE be also uniform ; in which case
P describes the spiral of Archimedes. The fluxion of SM is
to the 0uxion of y^ as the constant velocity of M is to the
constant velocity of N, or as EM is to the arch EN. If the

ratio

Y

i
#

\

X

I

i

>

Chap. VII. • Cfthc Tangents ofeuroc Lintn. igi

Tatio of those floxions be greater than that of SE to SD, lei it
be the same as that of SE to SA ; join E&, and^ since ED is the
tangent at E^ a part of EA^-mnst fall within the spiral. Let P be
a point in the spiral within the angle AED^ and let PL perpen-»
dicnlar on SE meet l£Jc in V. Then^ since the fluxion of SE isto
the fluxion of/E as SE is to Sit, or as EL is to VL, EM shall
be to the arch EN as EL is to VL. But EM is less tlian EL^
and the arch EN is greater than VL ; therefore the ratio of
EM to EN is less than the ratio of EL to VL. And these be-
ing contradictory, it follows, that the ratio of the fluxion of
SE to the fluxion of flS^ b not greater than that of SE to SD.
If the ratio of those fluxions be less than that of SE to SD, let
it be the same as that of SE to SK; produce DE and KE to d
and I; and from p a point of the spiral within the angle d£lf
let p/ perpendicular on SE.in /meet the .right line £1 in v ; let
Sp meet the circle /E6 in n, and Sm be equal to 1^. Then^
since the fluxion of SE is to the fluxion of/E as SE is to SK,
or as E/ is to ul, Em is to the arch En as E/ is to la. But Em it
greater than E/ ; En is less than its tangent Er, and therefore
18 less than lu; so that Em is to Eir in a greater ratio than £/
is to lu. And these being contradictory, it follows, that the
fluxion of SEis to the fluxion of the arch/E as SE is to SD.
From this it foQows, conversely, that if SE be to SD as the
Telocity of M in the ray SE is to the velocity of N in the cir-
cle yE&> or (supposing S^to be the position of the ray SN
when P sets out from S the beginning of the spiral) as the ray
SE is to the. arch /E, then shall ED be the tangent of the
spiral at E ; and this coincides with the 21st proposition of^
the treatise of Archimedes concerning that line.

1203. Case 2. Suppose now the motion of M in the ray SE to
be continually retarded while the motion of N in the arch NEn
is uniform. It follows, from the fourth axiom, that if the mo-
tion of M be continued uniformly from the term when it comes
to E, a less line than EM will be described by it in the same
time N describes an arch equal to EN ; and, the fluxions of
SE and /E being measured by the velocities of M and N at the
term when they come to £, it follows, that EM is to EN in a
greater r^tio than that of the flvxion of SE to the fluxion of

1

192 Of the Tangents of curv^Lktti. BookL

/E. If the mtio of those floxkms'be greater tlian that of 8E
to SD^ let it be thesame as that of SE to Sk, or that of £L to
LV ; and, P being a point of the curve within the angle AEDj
as in the last article^ EM shall be to EN in a' greater ratio than
£L is to LV ; which is impoasiblej because EM is less than EL,
and EN is greater than LV. If the fluxion of SE be to the
fliudon ofy^ in a less ratio than that of SE to SD, let their
ntio be that of SE to SK^ or (the conatmction being the $smm
as in the last article) that of E/ to &. It follows^ firom the
third axionij that Em is to En in this case in a less ratio thaa
that of the velocity of M to the velocity of N at the term when
these points come to E>or the ratio of the fluxion of SE to the
iuxion offE. Therefore Em is to En in a less ratio than E/
is to lu. But this is impossible^ because Em is greater than Ei,
and En is less than lu. Thorefere the fluxion of SE is to the
jBttxion of /E as SE is to SD.

904. CatcS. Letthemotionof M0/£g.58)intheraySEbeacoe-
lerated continually while the motion (^ N in the arch NEe is
unifiurm, but so that the arch PEp maybe still on the same side
of the tangent ED with S ; and the fluxion c^SE shall be to
the fluxion of/E in a greater ratio than that of EM to the arch
£N> by the second axiom^ but in a less ralio than that of £m
toEfiy by thefirstaxiom. Let Sa be perpendicular on the tan*
gentina; and^ if the ratio of those fluxions be that ofSEto
SK^ whidi is kss than the mtio ofEaioSa, it follows, from
the SOOdth article, that a right tine ST may be drawn fiom S
meeting the tangent ED in T, the drcle/Ei in N, and the
tangent of the arch EN in Z, so that TN may be to EZas SE
is to SK. Let this right line STmeet the curve FEinP; and
since TN is to the arch EN(which is less than the tangent EZ>
inagreater ratio than SE is to SK, and therefore in a greater
ratio than EM is to EN, it follows, that TN is greater than
EM or PN, and that the pmnt T is between P and S, against
the supposition. If the fluxbn of SE be to the fluxion of /E
as SE is to Si, then, by the 199th article, a line St may be
drawn from S meeting the tangent DEprodeeed beyond E in f,
and the circle /E6 in n, so that the ratio of te to dbie chord Vm
may be the same as that of SE to S^ or of the flexion of SE

t6

Chap. VII, CfOc Tat^^mt 9f twr^ UMi t93

to the fluxmi of /£^ whiob is lete than the -ratio of pn to
tJke oroh En. TlMvefdre> ttrpt>09mg S^ to be titcfa d Une^ tbs
ratio of In to the arch Edi shall b^ ksathaii the ratio of|iii to
tiia tame arch; add i«»ibattbe lesb thatipn> so that the ^foka^/t
diall M betwikt p ind S^ agaiiffit the siippositioii« Thertfode
the fliunoaaf SEiit to the fluxion Off /Bin a tslao that is nei*
tber greater ribr less than that of S£ to SD. '

^03. Cau4. LetthepaiDtSO%.^abdtfaeowrveFI^be<m
diffcient sides of tht tangent BD ; atid, the motite of N being
unilbdsi^ tbenotioil of M inufi be perpeiaalty aooeleratcd, by
the wist ifirtkie. in thitf case^ lhefins!i<)n of S£ i^ to tbeflux-
ion of/B k » gteatar ratid than that of EM ta the arch £N>
by the sewoAsuionif bot in a ktt ratio than that of Efl^ to the
jsrah £«> by the irst ajdoih. If the ratio of these fluxions be
that of S£ toany Kne SK grestar than 80, let Q be any pobit
betwixt D and K ; join EQ ; and, by the 2(X)dth artide, a
right line SR may be drawn from S to the right line £Q meet^
ing it in R, the circle/£6 in N, and its tangent in Z, so that
RN shall be to EZ as SE is to SK, and, consequently, in a
greater ratio than that of EM 16 the afch EN. Suppose there-
fore SR to be such a right line, and RN shs^U be greater than
EM OF P^ ; and^ because the ratio of RN to £Z continually
increases while N moves towa^ds^E^ it follows^ that while N
describes the arch N£, the ratio of RN to the aieh EN is
greater than that of SE to SK, and tbajt RN is greater than
PN. But RN is less than TN v therefora the right line RE
passes thvoag^ the angle of oontact PET formed by the curve
PE and ite tangent ET"; which is aboard, by art. 181. If the
dttxioQ^c^ SE was to tbo fluxion of/£ as. SE is' to any Ipne
S& les^ than SD y then, taking aay point q betwixt D and k^
jmningp f Ey and*]|^codiicing it beyond E^ a- light line 9r might
be dll'^awD te^ it(by art« 199), meeting it in r, and the circle £fr
kk ft, so* thaS m asight be to ibe ehoid Bm as SE is> to Sib, on
asithe HiifS^p^ci SE is to the fluxion of^E, aliid, coi^sequently^
in>akasra^tbianthatof£ffl^pSjpiiyto'tbearch£it. LetSrbe
soch a line^ and r^shall be^kss- than fu^ Suppose the point n
10 aov^ from n tkmards £| and the ratio (rffw^ to £i» shall do-
cvea^e ^ontinntilfy, by art ig9> wd shaUbete^thuMliatof SEto

VQL. I. N S;^;

1^ Of the Taf^enis rfeuroe Lina. Book t.

' SA ; therefore^ during that time^ m is less than jm ; but it b
^ glieater than tn ; sothat the right tine Er most pass through the
«angle of contact /?£/ formed by thecarTeEpanditstangentfify-
whtch is absurd^ by arU 181. Tbepefi^re the fluxion of S£ is
to the fluxion of/£ precisely as SB b to SD. By joining these
*C8ses together, the demonstration b applicable \f hen. the motion
of M is accelerated on one side of •£ and retarded.on the pdier ;
f8mll>y the ninth and eleventh theorems, and the 60fh article,
itb rendered general. . When the tangent ET'b perpendicu-
lar to SE, and the velocity of N b given, the vdocity of M at
-E and the fluxion of SEvanbhes. When the ray SB touches
,the curve in E, ^d. the velocity of M in the ray S£ is given^
the velocity of N at E and the fluxion of^E vaniidies. This
proposition may be demonstrated in another mamier ; but
thb seems to have the nearest affinity to the method of
Archimedes.

PROP. XVIL

!206. The same things being supposed as in the last prcpositio7i,
thefltution of the curve FE (fig. 5Q) is to theftuxion of the ray
SE as the tangent ED is to the ray SE;

For let SP meet the tangent always in T, and the velocities
of the points P and T shall become equal at the term when
they come together to E.

First, let the curve FP^be convex towards S,and SP increase
continually, as in the last article ; then, the motion of N being
supposed uniform, the motion of P in the curve shall be acce-
lerated perpetually, by art. 201. The motion of the point T
is abo accelerated, and its velocity at the term when it comes
to E is less than the velocity of P at any term after P passes E,
as when it comes to p ; because a less line thiui Ef would be de-
scribed by the motion of T continued uniformly firom ȣ,' in
the same time that P would describe a greater line than thettrcb
Ep (which exceeds ISJt) by its motion <:ontioned uniformly from

p, by

Chap. VIL Of the ToT^nts of curve Lines* 195 *

p, bjrthe first and second axioms. Therefore, if the velocity
of T at the term when it comes to £ was greater than the ve-
locity of P at that term, it might be equal to the velocity of
P at the term when it comes to some point h betwixt £ and|9.
Let Sh meet Fd ing ; and it follows from the second axioms
that the point P would describe a line greater than the arch £&
hy its motion continued uniformly from h, in the time it de»
scribes £A with its accelerated motion ; so that the point T
would describe a greater line than £g by its motion continued
uniformly from £ in the same time that it describes £g with its
accelerated motion, against the first axiom. In lik^ manner it
aj^ars, that the velocity of T at the term when it comes to
£ cannotl>e less than the velocity of P at the same term. There*
fore these velocities are equal, and, by prop. 15, the fluxion of
the curve FP is to the fluxion of the ray SE as ED is to SE.

207. When the point P describes a circle that passes through
the point S, the motion of N being supposed uniform, the mo-
tion of P is also uniform. In this case, P£ is always greater
than TE, andjoE less than fE. Therefore the velocities of P
and T are equal at the term when they come to E, by the 53d
article. When the point P describes any other curve that is
concave towards S, its motion is in some cases accelerated, and
in others retarded. But it follows, from what was shown in the
64th, 55th, and 56th articles, that the>e]ocities of P and T
are equal at the term when they come to E m those cases also.
Therefore, in general, the fluxion of the curve FP is to the
fluxion of the ray S£ as ED is to SE.

208. Angles are measured by the arches which subtend them
in equal circles ; and in general they are to each other* in the
ratio compounded of tlie direct ratio of the arches which sub-
tend them in any circles^ and the inverse ratio of the rays of
tliese circles. The angular motion of the ray SN (fg, 60)
that generates any angle ASN is as the velocity of N in the
circle EN when the ray SN is given, and is always as the ve-
locity of N directly, and the ray SN inversely. The fluxion
of tlie angle ASN is in the same ratio.

. 209. Let the right lines CP, SP re vol ve about the given points
C and S ; and let their intersection P describe any curve FEH

N 2 'in'

19^ Q^'^^ TsHgents ofiturvi Urnef. Book I.

10 the same plwe with tho^ poiiite ; produoe SC till it meet
the curve in E^ aad^ when P oomes to B> tiia ftiigtflar motioii oC
CP ^hall be to the angulajr iB^otion of 8P m SB is to C£. for
let the cirole £N descril^ from the centre S through B meet
SP in N ; and let the circle BR. dcsicribed from the centre G
tfiroagh £ meet CP m R ; and let SD, CB he perpeiidi<
A£ the tangent of the curve at B in X> and B. Then
lodty of N shall be to the velocity of P at the term wheii they
come together to B> as SD ia to S£. The veboity of k shall
be to th^ velocity of P at the same term> as CB is to CE ;
and 8D is to BE as OB is to CE. Therafgie the velo^ty of
N is equal to the velocity of R at that term ; and the aogiUar
motion ef SP is to the angular motion of CP, when P 001009
to £, as C£ is to SB^ by th^ last artioie.

PROP. XVIII.

SIO. Let P (jSg. 61 and 6a) the iMte9$ectum of the right Km^
CP and 8P revolving about the givimp^intM C and S^ d^
scribe the curve FEH; Ut AQ touch thit^cur^i im £; coas/t*
itite the angk SET e^ual to CEA^ so Oo/ ET and EA may
Kedifferetawa}fsfroniSl^audCl^\ mndhiKSmeeiCSm
T. Then, if CA, Sa be night lineigivm iuf0sihon, thejhix-^
ion of the angle ACF si4M he to the JhusioH ofaSF, wheu P
comes to IS,, as ST is to CT.

From the centres C and S describe through tile pomt E the
circles ER. and £N, meeting CP and SP in K anJl N respec*
tively ; and let CB and SD be perpendicular on the tangent
AO in B and D. Let CK parallel to BT meet SE, produced,
if necessary, in K ; and let EQ be perpendicular on CK in Q»
The angle QKE is equal to SET or CEB, by the oonstFUClkMi;
and therefore the triangle QKE is similar to CEB, so thatG£
is to CB as EK is to EQ. The triangles SED, CEQ are also
similar, and SB is to SD as CE is to EQ^ Bat when P and R

come

r

ClMp. VII. Of the 3Wi«aii^y tmHfc UhOi 197

g6IM to Ey ike velteilty of P ii to the ydooit/ bf R •» GE is to
CB (by pro^. I5aiid 17), thafeisi a^ £K is to EQ. The Vclooi^
Of P I* tdtbe vcjlocity of N at tbesom^ term as S£ is taSP,
ctr «i C£ id to BQ. TMr^ofe Ihfe velooily of R is to the velo^
oitj ^ N^ lihtk th^ teme topBiker to E) as CE is to EH.
This tetio com^imde^ with the inTofse ratio of the tty GR
to tbe ray SN, or with Ifa^ ratio of &E fo CE, gi^es tbe ratio
of S£ to EK, dt (beeaase ET is parallel to Gk> of ST t«
CT \ which fherdbre (by art. dDS> is the ratio of the aogulat
vekoeity of CP about C t6 the angular telooity of SP about S
wftoil P Gomes to E, or the paftio of the flOxion of the cfdgid
ACE to ihe fhnion of the angle <rSE. When the apgle CEA
is etfSBi 10 CSE, the point K faHs^ on S> Ae angular oliotions of
CE «ad SE are equai/ and tbo iftixiOn 6f the angte AGE is
equal to the AnifiOO of oSS. When the takq;BBft A£ passek
thsosgh S^ iAmi point T fells on C ; and in ihis cosej tibe angti>-
kor vekidty of the ray CE being given, that of the ray SE va«^
nisbes. The pmnt E may coincide with Sy but we i^rve this
baJe tin #e come to treat of the corvatore of liAes. The flux-*
wn o#theatigle GPS is equal to the scntior difleftece of the
9mfAa» of the angles ACP, irSP, and is to the flinioir d( the
angle AGP or the fluxion of ASP, when P cornea tb E, as CS
ktoSTortoCT. When the points C and SO^g.GS^ are ik>t in
the plaM of the curve FEH, the ratio of the asigtilar velocities
of thera^ C£ and SB may be disdoced from this pvopoaitton*
For if V bo any point in ^e tangent AB, aad the right Une
V/e€(tial to VS be drawn in the plan^ CVE oonstitnting the
angle /VE eqaul to^SVE, tlie anguter velocity 6f/£ about/
shdl he equal to the angtdar velocity of SEaboot E.

tU. The preceding propositions ihow the anaiogy tlfei^ is
betwixt the metfiod of Fhixions and the method of Tangents^
and sepvO for determiiiing the tangents from the fluxioBs of Unes
itadaogles^ as well as for finding, these fluxions from the tan-
gents. ]3esides these general theorems^ there are many parti*
cdor propositions drnt are often of use in deternrinikig the tan-
gents of curve hues ; some of wliidh we shoH briefly describe.
^ S (fig. 64) be a given point in the plane of the curve ALB>
JLP a tangeat at L any point of the ourv^^ SP perpendicular on

N 3 the

L.

• I

198 Cf^J^ Tangents iff curve Lines. .. Book L

the tangent in P; and let P be always found in the curve DPE.
Let PT constitute with SP an angle SPT equal to the angle
SLP^ on the same side of SP as LP is of SL ; and PT shall
be the tangent of the curve DPE at P. For^ firsts let the arch
U of the curve ALB be concave towards S, and the rays drawn
from S to the arch LI decrease from SL to S/^ and let Sp be
perpendicular on Ip the tangent at / in p ; join Yp, and htpl
meet PL in R : then because the angles SPR^ SpR are right,
the angle SPp is equal to SRp ; and the angle TPp is equal to
the difference betwixt the angles SLP knd SRp. The angle
6LP is equal to the difference of the angles SRP and RSL;
the angle SEp is equal to the difference of the angles SRP and
PRp or PSp : therefore the angle TPp is .equal to the differ*
ence of the angles PSp (or PRjp) and RSL. But, by supposing
/to move towards hy the angles PSp aiid RSL and their differ«»
ence may become leas tlian any given angle. Therefore, while
p moves towaords P, the angle TPp may become, less than any
given angle. From which it follows, that no right line can be
drawn through the angle of contact formed by the right line PT
and the arch Pp. Therefore PT is the tangent at P. When
ALB is a parabola, and S its focus, DPE is a right, line; but,
when ALB is any other curve, DPE is a curve line. In the
parabola, SRp is an angle always of the same magnitude where*
ever the point / be taken, the point L being given ; and PSp is
always equal to RSL, or one half of LSI. . In other cases, ac-
cording as the angle PSp is greater or less than RSL (or the
fluxion of the angle ASP is greater or less than one half of th^
fluxion of the angle ASL), the angle SPT is greater or less
than SPp, and the arch Pp is concave or convex towards S.

£12. When the arch JJ (fig, 65) is convex towards S, the
constniction in other respects being the same, producepP to p;
and the angle SPp shall be equal to 4SRp. Therefore the angle
TPp is equal to the difference of the angles SRp and SLP,
which in this case is equal to the sum of PRp, or PSp, and
RSL; and, since this sum may become less than any given
angle when / moves towards L, it follows, that PT is the tan-
gent of the curve DPE at P. In this case the arch Pp is con-
cave towards S. Let ST be peipendiCular on PT in T, and let

the

the point T be always Ibund in tbe cuhre FT ; tiiea the lan-«
gent of this curve at T ihkW cotistitute an anjgle with ST eqoil
to SPTk)r SLP. Tliere is a series of curves xtshlch^may be co&^
cej^ to be derived froiH' eaab other in tbismaaisec: the tan*
gents fonn always ec^al angfes tihh the rays drawn from S at
tbe corresptonding points <^' the curves, and the fluxions iof
the cuirves are to the fluxioi^s of the rays at the«; points in ihi
same ratio in them all.

dia. Let any cnrviBneal figure C£H(^. 66)be thebainajf
annp GMubal sarfade that hftaiis vertex in V^ ani£T beth^tadi
gent of the arch EH in £. Let cek be a sectibnof this coB^qal:
figure made by any {dane ; andlet etbe th&cobunon seotsoa oft
thatplaneand thej^lane V£ffi:[thenshaliie/be tfa^ctang^i^
the arch ei. For, if e/ meetithe arch eh in axkygqiitt Jaesidei r^V
is manifest that BT must meetlthe^arch EH issomeotber^Vebtn
becides E ; and if any right line, as €x, cna bcudrwibD thfonghf
the angle of contact kei formed by tbe arch e^and^ecigfat'ljhe^
e^, let the common sectioaoftlie planes Visx^G£U^'beBX;sdQdi
EX shall pass through the angle of contac4i UtBTySgaiiiatihAi
supposition. It is obvious, that the arch eh and its tangent et
are the shadows of the ^rch EH and'the tangent. £T formed
by rays issuing from V upon the plane ceh.

214. Lettheright lines Cti, SK(^g. 67) revolving aboutthe
poles Cand S by their intersection P describe the curve APS that
passes through S; and letSA: be th^ situation of the right Ijne
SK when CH passes througl;!. S ; then shall Sk be the tangent
atS.

CHAP. VIU.

Of the Fluxions of curve Surfaces.

I

V -

215. xjLRCHIMEDES establishes his theorems qoaoeming
curve surfaces upon this principle, Thaty when two curve sor*
faces have their concavities turned the same way, and haw
the same t^rms^ihat which includes the other is the greater s^*
face : and this axiom is sufficient for demonstrating the caae^
thai were considered by him. But, because it cannot be appUed

N4 for

^

foe roniMring sutf <ico» thut are ge«^^«»ed by omfen ifblsb •«»
tenr^x iowModf the wis ab^nl vbi^b Ihe figuf^ m ni|ii>(i9Qd ki
Kevolvi^ yre ahftll jn»k^ use of llw following ptincilple i« place of
i^thatisimone geiieial,aiidfiMii4tQbeQo]aii^vido^ 1M
GBHh^ aoLjr axch of a eiirve ; Md tet BG the iiw be divided by
i coDiiauti bisebtion ioto any number of eq^^ pnits BK^ KD>
DI>.:LQ. 1^ the mdiQfttes fiC. KM^ P£i I^N, GH (Jig. «ft

n. 1, 2) ; meet the curve in the points C^M, fi, N, Htud lei Um
tei0B&t%al; C^ M, Ei H foim the eircuinaenbedfiguie CQRaOH.
Then» l^r coatiinuDg to bisect the pacts of the exit SG^ wA
au^pceingthe enUDates^ (^ords; and tasgeats to be drair n aa be«
Ave, the petimeteraof the oiroumacribed and mfcribed figurea
eOB£QH, GMiNEH ahaU approach ocmtiBually to ftbe aroh
(S^/H ; AndJiupposiiQg the whole figure to revolve aboat the
ttdiL^G, the soifacQs desccibed by those periaoetera shall ap-*
prpaoh IQ tfaesurfhcBdesoribed by the arch CBH» sfx that the
chii|3rfii)cea ttefarixt them may heeotiifi less than any ffvtn aor^
fbee. . TbaiakSdiD <aited from. ArckkmBda ia art. l%d, stay be
deduoad A'^ip thh ^ikicipie.

• ' ., • ,

U.M .': 1

,;1. .„,,, LEMMA Vm.

Slt^. Let ijje' kgkftine flC (fg- Gg) produced meet the axis

*'\fi^'inA', aiid tHe surface described by HQ rttotoing about

AG shall be equal to the area of a circle the radius of which

is a mean proportional befrdxt .?HC and the right line DE

tftat bisects HC in E; and is perpendicular to the axis in D.

Let HG and CB perpendicular to the axis meet it in G and
B. By what waa shown after Archimedes, in the Introdijiction^
page 4 V, %he soFfboe described by AH is equal to a oirck the ra*
dins of which is a meas proportioiial betwixt AH and HG; and
Uie surfcce described by AC is equal to the area of 'a oiiele thai
has its radius a mean proportional betwixt AC iusd BC. There-
fore theee surfaces are to each other as the rectangle AHG is to
ihe rp<5taDgle ACB, or as the square of AH is to the square of

AC ;

lUevn/txAv^wy.

r<

% •

r

Cbtp. VIII. Of the FbixtoM if curve Surfacis. fiOl

AG; 9Ad the surface deaciibed by CH i» to the surface describe
td by AHj as the (tUfoenoe of the squares of AH and AC is
I to thf s<iQaie of AH> that is (Elem. 8. d.) as the rectangle goq^

tabad by 4AE and £H is to the sqaare of AH^ or as the rect*
angle contained by 2CH and DE is to the rectan^^e AHG«
But the area of a circle whose radius is a mean proportional
betirixt sCH and D£ is to the surface described by AH in
the same ratio. Thecefore the surface decribed by CH is equal
to the area of a circle vhose radius is a mean proportional
betwixt £CH and DE. This coincides with the l6tb pcoposi-
tion of Jrehmedc^B tareatise cono^ming the spheare and cy«
liuder.

. ei7. C#r.I. When the sods o£ the cone AB increase am*
fidrnily^ the convex surface of the oom^ described by the right
lim AC increases with a motion that iacoi^tinually accelerated ;
ht vhaft B6 the incveoient of tiie axis ia given> the suiiacede-
scribed by GH (which is the simultaneous incremient of the co-*
meal surf aca) is aa D£i, which imaeaaea in the same proportion
aa Al> incsea^.

eiic Car. U. Let tk# right Ikkm KC« HK meet BC that i^
pevpeadiculnr to the 9pmT8Gs in C and K, aad let them meet
DE in ]^ and L. Let HN bAseot ihs angle CHK^i and HI per-
pendiralar to HN sbaUjMset DE produced beyopd E when tb&
aiiffeGUNislesathain^rjghtwe; k( them meet in L Then^
the figura being su^p^sed Wi«vo1vq about the axis BGj thesur*
face desefibed hy ISC ^sAbtv%> th^siirfttoe described by HK
as the Kctongte D£I is %» Iherreot^l^ DU. For let LE per-
pendbcnlarto HN wmi WSi ii% R ; andoithe a^ig^ RUl being
equal to LRH> HL shall be equal to HR ;. and EH shall be:
to LH (or RH) «a £1 iate^ L[;so that the reatangk DEH
diall; he to.the veeteP^ DLH as D£I is to DXL But fay this
ltmni0, the surface described by HC ia to the surface described
by HK as the neelaiagie D£H is to* tbua rectangle DLH, and>
ooBsaquently^ aathe reoMgle DEi is to DLL

iilQ. Cor. UL O^, 7<lb)Ilft>Uows from tibe last corollaiy,.

that when D£ ia equmUe L]> Hie sulfates descdbed by HC and

' HK areequal. Therefoie, if 6H beproduoed beyond H till HM.

be equal to GH, AF parallel to GH meet HK in Fji EM meet

GO

202 Of^ the Fluxiom of curve Surfaces. Book \,

GO perpendicular to HNin O^ and BC produced pass through

; the surfaces described by HC and H K shall be equal; For
in this case^ DE^ HI> and OM intersect each other in die point

1 that is middle betwixt O and M ; and^ because GH is equal
to HM^ DE is equal to LL

220. Cor. IV, Let CV (^g. 71) be perpendicular on HK in
V ; produce it till it meet DE in S ; and^ ^en £S is either equal
to ED, or less than it, the surface described by HC is greater
than the surface described by HK. For; in the triangles £CS^
EHI, theside ECis equs^l toEH^the angle CES is equal toHEL
but the angle ECS is greater than EHI (because the angle
ECS is equal to the right angle CVH added to CHV, where-
as EHI is only equal to the right angle NHI added to CHN) ;
and therefore ES is grefafeer than EL Therefore, when £S is
either equal to ED, or less than it, DE is greater than £1, the
Rectangle DEI is greater than DLI, and the surface described
by HC is greater than the surface described by HK.

221. Cor. V. The taoae things being supposed as in the latt'
corollary, let k be any point betwixt C and. K, and the sur-
fece described by HC shall be greater than the surface describ-
ed by H/r, which is itself greater lllaft that which is dcscrib--
ed by HK. For let Cii perpendicukt 6n Hfe in u be produce
cd till it meet DE in /, and, ff being less* than ES (wliich^
is supposed to be either equal %b ED, ^ less), it follows, thal^
the siuface described by HC is greater than that which is de*
scribed by H£. Let HA meet EL in e, and ko perpendicular
on HK meet ^D in f, kxA eS bmUg less than eD, the surface
described by HA must be greater than that which is described
by HK, by the last corollairy.

222. Cor. VI. Lei A (Jig. 7^) be apoint on the line HC be-
twixt H and C, or any where within the triangle CHK; let
AA parallel to HK meet CK in A, and he meet CK in any point
c betwixt C and A ; then the surface described by ho shall be '
greater than that which is described by AA. For let hg paral-
lel to HG meet the axis in A, let de parallei to hg bisect 'Rg in
J, and meet Ac, AA in e and / ; and it is manifest, that if cu be
perpendicular to AA, or parallel to CV, it shall meet ed in some
point / betwixt e and d.

223. C-or.

.Cb'ap.A^IIl. Of tke Uuxiom of curve Surfaces. fi03

. d£S. Cor, Vll. It appears io the same, manner^ that, whea
h (fig. 73) is any point within the triangle CHK, c a point on
CK, k a point on cK> if ch and kh produced beyond k meel
CH in r and n, so that the angles Crhy Cnh be each less thaa
the angle CHK, then the surface described by he shall be great-
er £ban the surface described by hk.

. S24. Cor. VIII. Let HEC (^. . 68, n. l)be now aa arch
of a continued curvature convex towards the axis BG; let tha
ordinates from BC to QH increase continually, and let iba
-ordinate DE bisect BG; let HT the tangent at H meet BC in
T, and suppose that Hv perpendicular on CQ the tangent at
C meets DEiny^ so that j^is either eqioal to th^oidinate D£^
or less than it. Then, the surface described by the arch H£C
revolving about the axis BG shall be l^s than the surface de-
scribed . by the tangent HT revolving about the same axis*
For let HA parallel to the tangent CQ meet BC in i, and T«
in V, the angle TVH being a right one, and the arch HEC
being within the triangle TH^, it appears, tliat, if the chord
HE be produced till it meet jCTT in r, the surface described by
HT shadl be greater than that which is described by lAr, by
~ cor. 5, and the surface described by Er greater than that
which .is described by EC, by cor. 7. Therefor^ the surf^cede-
acribed by the tangent HT is greater than the sum of the sur-
faces which are described by the chords HE and EC. In like
manner, the parts of the axis BD, D6 being bisected by the
X>tdinates KM, LN, if the chords HN, N£, EM, be produced
till they meet CT in u, x, and z, th^ surfaces described by Nu ,
Er, Mz, shall exceed the surfaces described by Njt, Ea, MC
re^ctively, by cor 7, and therefore the surface desgribed by
HT being greater than that which is described by Hti, which
exceeds that described by HNx ; and this surface being greater
tliaa that which is described by HNEz, which exceeds the sur-
face described by HNEMC,it follows, that the surface describ-
ed by HT is gres^t^r than that which is desqribed by HNEMC.
It) general it appears, that the surface described by the tangent
HT is greater than the surface which is described by the peri-
meter of any rectilineal figure inscribed in the arch HEC ;
a^d it may be shown in the same manner, that it is greater
thim the surface described by tibe perimeter of any rectilineal

figure

004 Of tie Fburiom of curve Surfaces. Book I.

figure droniMcribfld about the arch fontted by its tangettts*
Therefi>re Ae sorfiice descfibed by the tangent HT Is greater
than that which is described by the arch HEC^ fay articfe

M5. Cor. IX. Join 6C, and, if the aaogfe GCQ irkkh it
forms with the tangent atC be eitiier a r^ht aag^e^ or greater
dMtt arigtA^^ne, and TBbe Mot lesB than TC, the nufate de-
•cribed by the tangeivt HT shatt be gretfto than that wfaioh it
. iteicribed by the arch UEC : for, in cbig ease, tf n dther
e<(iial fo^EB^ or leM Aimi it* It appears, therefore, thait^ wliea
(the tuxve i^^ortv^t ttiwaids the «%ia, and the ordinates increase
«iM«lhe wm meteum^r^iA arch of the carve, as CH, may be
iaiea of a finite magnittide, so that th^ snt&ce described by it
shall be tM^ Iham the sariaoe which is described by HT tile
Saagent at H term ineted m T by the ontosafo from C. If the
tangent CQ meet GH in Q, it is evident, that the surface de-
■cvibed by CMENH, the perimeter of aivy rectiiineat . figum
inscribed in the arch CEH is always greater than the surface
described by the tangent €Q. Therefore the smface descdb*-
ed by the arch CEH exceeds that which '» described by the
tangent CQ> hj art. £15«

£26. Cor. X. Let i4ie cur^ F£H be convex towards the
axis AG, and the ordinates ptn increase while the axis Ap in^
creases ; thffl, if the axis Ap increaae anifoiad}y, the surface
described by llie curre Tm shall increase with an accelerattd
motioB. For, while tlie axis acquires the equal augments BD;
SG, let ^e curv^ acquire the angmentsCM £ and BNH ; and
it is manifest, diat the surface described by the perimeter of amy
tectitmeat figure inscribed in the arch HNE r»gt«ater than ^
surfftce described by the perimeter of a rectilineai figure in-
scribed by a similar confiftruction in the ansh EMC. Therefore
the siNpface described' by the arch £H is greater than diat
which is described by the arch C£. Let CQ meet Dfi in q,
and^ the tangent at H meet DE in I ; then, the surfoce de-*
scribed- by the arch CE being greater than that which is de-
scribed by the tangent Cq, and the surface described by the
arch £H being- less than that which is described by the tan*»
gent N#when CEH is dimini^ed, as in cor. g,it foUows, that

the

Chap. VIII. Of tht JFInmHtf tfewv€ Sitrfaca. <05

t}ie motion with which the snrfiu^e described by Fm Aowb^
while m describes the sareh C£H> it sjccderated in a conttniied
manner.

M7. Cor, XI. {fig. 75). The rest remaining as in the lul
fx>roUgry^ when the arch C£H is concave towards the axis
B6^ BD may be taken so small that the sorfiaoe described by
the tangent ET shall be leas than the surface described by the
chord C£ (by cor. 4% and therefore less than the surface de»
scribed by the arch CE, whichexeceds that which ia described
by the chord CE, by art. 215. The surface described by the
arch EH is less than that which is described by the tangent
£/^ by the same article. In this case^ when the axis increasea
nnifonnly^ the motion with whloli the surface described by
the curve flows may be uniform^ accelerated^ or retarded.

LEMMA IX.

$28. X#( BE and Qt (j%. l^fcrfendkular to the axis. aG
meet aQ j/e £ md I ; amd^ in ti^ <amfi tim& that the m^on
mith which the axis oD flows continued uniformly generates
D6^ let ike motion with wkkh the conical mff ace deuriied
by oE flows conHtmedumformfy genepat^ a space equal to R ;
then shall the space R be efual to the area of a circle the
radim of which is a mean proportional betwist D£ and I^EL

Tot, if the spaee R be gnsaler than saeh a circle lefe Ad bi
greater than ADin the aaae raftki; 9n^ if d& parald to DB
mee* £^ in o^ R Aall be eqnal to a eitck whoae radiaa i» a
sMan prc^MMtioaAl betwixt da aad it&; hecaaaethe area of
thia cirdk ia to the area of a cirele whose radiaa is a. mean
projportaoD^ betwixt BEaad iBt aa do iato I>fi> or as Ad ia
to AD. The space which is generated by the motion with
whidi the surface oEe flowa continued amformly> m the time
that the axia al> by flowing nmfermly acquires the augment
B4 is to the space R aa DG is: to Dd (by theor. d)> or aa
£^ is ta Ea» and therefore ie equal tntheareftof a ctrdewhose
Tidins is a mean propavtinoal hetoixt. da aad S£a ; bnl

the

^pa Of the Fhunons of curve Surfaces. Book L

the area of this circle is greater than the snrfiace described by
the right line Eo, by lemma 8> and the conical surface d£x ia-
ereases with an accelerated motion when the axis incieHses
oniformly^ by art. 217* Therefore a greater space is generat-
ed in the same time by a motion continued uniformly^ than
when the same motion is continually accelerated, against the
first axiom. In the same manner it appears, from the second
axiomy that the qmoe R is not less than the area of a circle
' the radius of which is a mean proportional betwixt D£ and

PROP. XIX.

229. Let DE and GH (fig. 74) perpendicular to the axis meet
the curve in E and H, and let GH meet the tangent at E in
i; then, the fluxion of the axis being represented by DG, the
fluxion of the surface described by the curve FE shall be
accurately measured by the area of a circle the radius of
which is a mean proportional betwixt DE and 2&.

Let the ordinate PM always meet the curve in M and the
tangent aE in N; and the motions with which the surfaces
F/mM> a^n flow shall be equal at the term when M and N
come together to E, or the fluxion of th^ surface VfeE shall be
equal to the fluxion of the surface oEe. For, the construction
being the same as in the^26th and 228th articles, first let the
arch CEH be convex towards the axis BG, and the surface
F/mM shall flow with an accelerated motion while P and M
describe BG and CH, by art. 226. Suppose, first, BD to be
so small, according to the 225th article, that the perpendicular
trom T upon the tangent at C may meet DE in D, or in some
point betwixt D and E, and the surfaceidescribed by .ET shall
be greater, than the surface described by the arch EC. But the
latter surface is greater than the space which would be gene-
rated by the motion with which the surface VfcC flows conti-
nued uniformly, in the time P describes BD with an uniform
motion (by the fiirst axiom) ; and the former is less than the

space

.Chap. VIII. Of 'the Fhscions ofcurot Surfaces. . ^ai

q»ace which woald be generated in att equal tipie by the motioa
with which the surface aEe flows contii^ued unifonulyj by the
second axiom. Thereft»e the motion 'with which the surfiaoe
aEe flows is greater than that with which the surface l^fcC
flows ;. and it is manifest from the same axioms, that it is less
than the motion with which the surface f]AH flows ; so that
it must be equal to the motion with which the surface ¥fMli
flows when M comes to some intermediate point of the arch
CH. If it be said to be equal to the motion with which the
surface F^xX flows, X being some point betwixt £ and H, let
4ZX perpendicular to the axis in d meetthetangent £r in o. The
motion with which the surface VfxK flows continued uniformly
generates a space greater than the surface EexX in the time P
describes JM, by ax. 1 . The surface EexX is greater than the
surface desoribed by the right line £o revolving about Hd (by
art. £25). Therefore the motion with which the surface a£e
flows continued uniformly, would generate a ^ace greater than
the surface described by Eo,in the time P describes D^f uniform-
ly ; but a less space than this surface would be described in that
.time by the motion with which a£e flows continued uniformly,
by the first axiom ; and these are contradictory. Therefore the
motion with which the surface oEt flows is not greater than
tlie motion with which the surface F/^E flows. Nor can it be
less. For, if it be said to be equal to the motbn with which
the surface VfyY flows, Y being any point of the curve betwixt
£ and C, let Yd meet the axis in d, and the tangent £T in o.
Then, the surface described by £o about the axis BD being
greater than the surface £eyY (by art. 2£5), and this surface
being greater than the space which would be generated by the
motion with which ^^Y flows continued uniformly in the time
P describes dD uniformly (by ax. 1), it follows, that the space
which would be generated in that time by the motion with
which a£f flows is less than the surface described by £o. But
it is jpeater than that surface, by the second axiom ; and these
are contradictory. There£Dre the motions with which the sur-
faces F/e£, a£f flow are equal. By the last lemma, the fluxion
of the axis AD being represented by DG, the fluxion, of the
surface aBc is measured by the area of a circle the radius of
i which

096 0/th^ BmiwAofauru Surfam. Bode I«

which bameaopioportaoiialbetwiictD£aiidd&. LeiDGbe
increased in anj pfoportian, and El shall be tncrefifled in the
tame proportion; so that» if D6 lepvenenk the fluxion of AD,
the flnxion^f the mnface TfeU shall be accurately measured
b; the aiea of a circle the radius of which is a meia pnqpoi^
tiooal betwixt DE and £&.

%Sa. Let the arch C£H0^.75)be now concave towaxda the
axis; andBD maybe takenin this ease so small tbatthesurfisce
described by the arch C£ shaK be greater than the surface dd^
scribed by the tangent ET(by art.S27); but the surface describ-
ed by Elis always greafaerthanthat whichisdeseribed by dieardi
EH. WheatlM:axbincxeasesanifb«mly,thenK)«io]iwithwbM^
tfaecofiiral surJacedesci ibed by aNflowsisoeMitinuidly aeseharaU
ed^ by art. %ig, aad^wl^iber the motion with wUds the nmhct
f^MLBomB be uniform^ accelesated> or retusded^it foUows^fima
what was denMnstialed in the 53d> 54th and 56th artidai^ that
Che motk—i with which the Bm&ice»TfcE,dEe flow are eqnsl
Therefave^ the flnxiojDi of the axis AD behsg iqpfeseBted by IK}>
the fluxion of the aadaoe F^Eis measured by the area of a
drde whose radios is a mean proportional betwixt DE and £E^
XbedmnonstxatknLk rendcsedgeneralby thepsopoaitiooB which
ham been so «Aen cited from the fiist cbE^ter.

25 1. Cat. I. LetEO(^|f.76)petpeiKiiculflar tothetaugcsitE^
meet the axis m O; astd^ because £0 is «» DE as El is td DOy it
feBowB, that the flnxioii of the suc&ce 1^;^ is nfesaBured by the
arenof a cirejbs wbeae CMhudsa meaapsopostianal betwiitfbftEO
and DG, whai the flaiiixxi of the axil is iqisesenicd by D6.
^Fhecefoser when F£H iaaftarch of a ciacfe whose draoaefeet
JkB coincides with die axis of motwi^diepefpcndietdar EObe*
ing infiffiable;,. the spherical socfacttAE^IbwsnBt&tndywfa^
the ax» AD flapv9 nnifonnly f and,i» the same time that the
uniform motion with whidhi the axis flows geaeratea AD^ tfsr
nniforminotioawillswhixdi thesphericalsariiacefto^vsgeneraacs
m surface e<|ual to the asen of a eifcic whose nHEus ia »niean
ppoportionai betwixt AD and AB; thai is^ thesphcskaisnsfiHsr
AEr is eqaak to the aseaof a cirde described with the dkosd
AE ; tfod the wbole surface of the sphcse iseqiial ta a circle
whose radiaa is AB the diametcg of the lyhirw, and thesefcae

) is

Cfaap.VIIL O/theFhudomofcyrocSurfaca. ^

18 qaadniple of a great circle of the q^ere, as was demonslrated
after Archimedes in the Introdaction.
iiS%.Cor. II. Butif the circle(y^.76)be mipposed to revolve about
a diord AB that is not a diameter^ let VK perpendicular firooi
the centre V upon AB meet it in K ; let L/ be the diameter pa-
rallel to AB^ Aa and Bi perpendicular to 12 meet the circum*
ferenoe in a and b ; produce D£ till it meet the circumference
again ine^andktitmeetARthetangiaitat AinR. Suppose
the right Kne R to be the radius of a circle whose area is A.
Then if R be a mean proportional betwixt fVK and the ex-
cess of AR above the arch A£ when £ is taken upon the lesser
ar<^ subtended by AB, the area A shall be equal to the surface
described by the arch A£ by revolving about the axis AB ; and^
if R be a mean proportional betwixt £¥ K and the sum of the
right line AR added to the arch A£ or at, the area A shall be
equal to the surfiEioe described by the arCh ae about AB. But
if £ be taken betwixt A ^nd L ; then, according as fl is a
mean proportional betwixt 2VK and the excess of the arch AE
above AR, or betwixt SVK and the sum of A£ and AR, the
area A is equal to the surface described by A£, or that which
IS described by ae. For, let 6/ jHToduoed meet AR in r, and
let £emeet L/in J; and, dnce Rr is to B6 as AR is to AD,
or VA to VK, and DG is to £f as rfE is to VE, it follows^
that Rr is to E^ as <flS is to VK ; and that VK is to D£ as &
is to the difference of Rr and Ef , but that VK is to J)e as £^ is
to the sum of Rr and Ef . Therefore the fluxion of the siirface
described by AE is measured 1l>y a circle whose radius is a mean
proportional betwixt sVK and the difierence of Rr and Ef, and
the fluxion of the surface described by the arch <ie is measured
by a circle whose radius is a mean proportional betwixt SVK
and the sum of Rr and Ef. But Rr measures the fluxion of
AR, and & measures the fluxion of the arch A£ or (U. There-
fore, according as R is a mean proportional betwixt sVK and
thfe diffetence of AR and AE, or betwixt SVK and the sum of
AR and AE, the area A is equal to the surface described by
the arch AE, orthat which is described hyae. When E is up«
on the lesser arch' subtended by the chord AB, the sum of the
anrfaces described l^AEand at is equal to the area of a circla
. VOL.1. O whose

210 Of the Fluxiom of curve Surfaces. Book L

whose-fVMliuf is a mean proportional belnfixi dVK and 2AR9 or
betwixt the diameter and 2AD ; but when E is betwixt A and
thy the sum of the surfaces described by A£ and ae is equal to 9
jdrcle whose radius is a mean proportional betwixt 4VK and
4he arch AE. Let bB produced meet AR in M^ and the sur*
iaoe described by the ^ole circumference of the circle^ by re-
aving about the chord AB^ b equal to a circle the radius of
fwhich is a mean proportional betwixt 4VK and the sum of the
jright line AM and arch ALa. The difference of the surfaces
^described by the arches A116B and AEB is equal to a circle
^rhoseradius is a mean proportional betwixt ISV K and the whole
fcircumference. WbentheaxisAB(^g.77)doesnotmeettheinr*
.de, let li be thediameter parallel to AB^and let LA^/B^ VK^ dD
be perpendicular to the axis in A, B, K^and D. Let Ebe in the
Aipper, and e in the lower semicircle ; and^ according as the
i^aare of R is eqnal to the sum or difference of the rectangles
contained by £VK and the arch LE, and by 2VL and AD^ the
aiea A is equal to the surface described by LE^ or that which
:is described by Le. The surface described by the arch ELeis
equal to a circle the radius of which is a mean proportional be-
.twixt this ai'ch and^V K ; and the surface described by the whole
.circumference is equal to a circle the radius of which is a mean
j>roportional betwixt the circumference and SVK. When the
axis AB touches the circle, this radius is a mean proportional
.betwixt the circumference and diameter.
t ass. Cor. UL Ingeneral,letPy'C^g,78,n.l,a)beanarchofany
-cwrre line, let ab parallel to AB meet DE in d ; and the areaof a
circle whose radius is a mean proportional betwixt iDd and the
arch F/* shall be equal to the sum of the surface described by
the arch Ff revolving about the axis AB added to the suriace
jdescribed by the same arch revolving about ab, when the arch
JP/'is betwixt AB and ab, but equal to the difference of those
tarfaces when AB and ab are on the same side of theardi If;
J>ecan8e> in the feonercasej the rectangle contained by ltd and
E^ is equal to thesum of the rectangles DEi, dEt, and in the
latter pase it is equal to the difference of those rectangles : and
therefore the sum of the fluxion of the surface described by FE
•bout AB^ added to the jSuxion of the surface described by F£

^

1

I •

1 »

I

' I

Cbap. Vjn. Of the Huxians of curve Sutfacd. 21 1

about ab in the fenner case^and the difFerence of those flaxioiis
in the latter case^ is equal to the ftS]|iton of an area that is al« .
wayf equal to a circle whose radius is a mean proportional be*
twizt 2Dd and the arch F£ : from which ii^foll<lws (by theon *
8)^ that this circle is equal to the sum of the surfaces described
by FE about AB and ab in the first case^ and to their difierence
in the second csbe. /

234. Cor.IV. C/;g.79)ItfollowsfromthelastcoroIlaiy^that^if
the axis ab meet the arch F^ in a> so that the part ¥a be on the
same side of ni with the axis AB^ and af be on the opposite
side of ab ; then a circle whose radius is ^ mean proportional
betwixt 2Dd and the arch Ff shall be equal to the surface de-
scribed by F/* about AB^ added to the excess of the surface de-
scribed by Fa about ab above that which is described by t^
about the same axis ab. Therefore, if the axis ab cut the arch
Ff in such a manner in the point a, that the surfaces described
by Fa, af (the parts of Ff that are on opposite sides of aft) about
the axis aft be equal, then the surface described by F^ about
AB shall be equal to a cirde whose radius is a mean propor-
tional betwixt 2J)d and the arch Tf And, converady, if the
sarface described by F/*about AB be equal to such a circle, the
surfaces described by the parts of F/* that are on difierent sides

^ of aft must be equal to each other.

235. Cor. V. Let AB(/g. 80) meetthearchF/inA,so that the
surfaces described by FA, A^(the pai;ts of F/'that are on dif-
ferent sides of AB) revolving about AB may be equal to each
other ; let aft perpendicular to AB meet the arch F/* in a, so
that the surfaces described by Fa, af (the parts of Ff that are
on diflerefit sides of aft) about aft may be also equal to each
other; and let aft meet AB in C. Then, if the same arch Ff
revolve about any axis CN that passes through this intersection
C, the surfaces described by FN, N/'(the parts of F/*that are
on different sides of this axis CN) shall be also equal to each
other. For, let CN meet the arch F/*in N; let K beany
point given upon CN, and KI be perpendicular on AB in L
Let E be any point of the arch Ff and let ED, EP,EM be
perpendicular on KN, aft, AB in D, P, and M respectively ;
and let EP (produced, if necessary) meet KN in R. The right

file C^Hk Fluxions afcumc SMrfacet. Book L

line ERb Ihe difisrence of EP and PR when tlui poiat Eis tak-
en upon the arch FA or af, in the case repiesonled by fig. 8(>«
and £R is the sum of EP and PR when E is taken, npon the
arch Aa. Bat ED is always to ER as KI is to CK ; aadj since
PR ia to CP (or EM) jas CI is to KI^ it foUowi^ that the rect-
angle contained by CK and DE is equal to the difiSerence or sum
of the rectangle contained by KI and PE^and that which iscon-
tained by CI and ME ; and^ the fluxion of the arqh AE being
rei»:e8ented by Bt, the solid contained by CK and therectangle
DEt is equal to the difference or sum of the solid contained by
KI and the rectangle PEt, and Ihat which is contained by CI
and the rectangle MEty according as E is upon the arches ¥a»
of, or upon Aa. Hence the solid contained by CK and the sur-
face described by the arch AE (or aE) revolving about the axis
KNj is equal to the difference of the solid that is contained by
KI and the surface described by AE (or aE) abontaft and the
solid contained by CI and the surface described by A£ (olt aE)
about K&j when E is taken upon the arch FA or af^ bat to
the sum of those solids when E is taken upon the arch Aa :
therefore the solid contained by CK and the surface described
by the arch FN about the axis KN is equal to the difference
c^* two solids^ the first of which is that contained by KI and
the difference of the surfaces described by Fa and aH about the
a:ds ab; the second is the solid contained by CI and the difier-
ence of the surfaces described by FA and AN about the axis
AB. And^ in Uke manner^ the solid contained by CK and the
surface described by the arch N/* about ELN is equal to the ex-
cess of the solid contained by KI and the surface described by
N^about 4tb above tbesoUd contained by CI and thesurface de-
scribed by Hf about AB. But it follows from the supposition^
that the difference of the surfaces described by the arches Fa
and aN about ai is equal to the surface described by the arch
M/* about the same axis ab ; and that the difference of the sur-
faces described by FA and AN about AB is equal to Uie surface
described by N/" about AB, Therefore the surfaces described
by the arches FN^ Nf about the axis KN ace equal to each
other when the surfaces described by FA and A/* about ABare
equal and the surfaces described by Fa and of about ah are

equal

rr-

Ghiqp* Tin. Cfikc Tlktimt of curve Surfuta. tlS

equal at the same time. This demonstration is easily applied
to the other cases that are not represented in fig. 80.

236. Cor. VI. It follows from what has been demonstrated^
that^ any arch Vf being given^ there is a certain point Cin the
plane of this arch, through which any right line KN being
drown meeting the arch in H, the mirfaoei described byFN^ jy^
(the parts of tibe arch that are on opposite side! of KN) retolv-
ing abocrt the axis KN iare always eqaal to each other. Hiis
point is called the centre of Gravity of the arch(beetol^^ if it be
supposed to consist of matter that is acted uponby an uniform gra-
vis in parallel lines^ the momentum of the part FN abbut any
aadsKNtbatpadites through Ci^equal to the momentum of thebart
N^about the same Bxhy as we may haVe occasion to show after-'
wards) ; and it follows from the two kist corollaries^ that the
soriace descrB>6d by the arch F/* revolving about kn, perpendi-
cular to any fine Cc drawn through C, Is ^qual to the area ot
a circle whose radius is a mean proportional betwixt sCc and
the arch PjTwhen Ten does not cut the atch F/*: but the diifer-*
ence of the surfiices described by the pdrts of the arch Vf that
are on oppbsite sides of kn is equal to that circle when kn cut^
the arch Ff. The area of such a circle is equal to the rectangle
^ohtained by the arch Vfand the circumference of the circle
described by the point C when the figure is supposed to revolve
about An; and therefore this corollary agrees with the celebrat-
ed theorem commonly ascribed to Guldinun.

237. Cdr. Vlll. Hence the distanceofthe point C from any
right line kn is determined^ when the length of the arch F/'and
the surface described by this arch revolving about kn are given.
For a third proportional to the length of the arch Tf and the
radius of a cfarcle that is equal to that surface shall be double of
the distance of C from kn . Thus^ because the surface described
by the arch AEB (fig.7^) revolving about the diameter U parallel
to the chord AB, is equ&l to thearea of acircle whoseradiusis a
mean proportional betwixt the diameter L/ and chord AB^ it
ibHows^ that the distance of the centre of gravity of the arch
AEB from the centre of the circle is to the radius as the chord
AB is to th& axch'AEB.

3 CHAP,

a

14 Of th^ grtnitxt an4 kast Of&wtes. Boqk I.

CHAP. IX.

Of the greatest, and kast Ordinata, of the Points of contrary
Flexion and Reflexion of various Kinds, and of other Jfec--'
iions of Curves that are d^ned by a common or by ajluxional
Equatiotf.

238. X HERE are hardly any speculations in Geometry move
useful or more entertaining* tlian those which relate to the
Maximd and Minin^a. Several prqpositions of this nature
are to be found in tl)e writings of ^e antient Geometricians ;
but they do np); seem to have had a general metl^ fpr resolv-^
ing prpblems of this kind^ Amongst the various improvements
that began to appear in the h^her parts of Geonietxy about a
hundred years ago^ Mr. de Fermat proposed a method for find-
ing the maxima and minima. How the methods that were
then invented for the mensuration of figures and drawing tan-
gents to curves are comprehended and improved by the me-
thod of Fluxions^ may be understood from ^hat has been fil-
ready demonstrated. A general way of resolving questiqqs cojfh
ceming the maxima and minima is also derived from it^ that is
so easy and expeditious in the most common cases^ and is so
successful when the question is of a higher degree> where the
difiiculty is greater, and other methods fail us, that this is just-
ly esteemed one of the most admirable applications of Fluxions.
239. When the nature of a variable quantity is such, that it
either increases continually without end, or decreases till it ya*
nishes,its greatest or leastmagnitudeis notas^gnable; and there
is no place for enquiries of this nature. But, when there is a
certain limit which the increase or decrease of the variable quan-
tity cannot pass, and the term is assignable when it arrives at
this limit ; or, more generally, when for some time Uie variable
quantity first increases till a certain assignable term^ and theii

* ApoU. Perf^. Conic, lib. 5. pni-f.

^ , cjecreascs.

Chap. IX. Of the graUeit and least Ordinates. 215

decreases, or first decreases till such a term, and then increases :
its magniUide at that term is considered as a maximum, or nnni*^
mum, without r^ard to its variations in other parts of the time.'

240. In the problems of this kind of the first degree, the va-
riable quantity is represented by an ordinate of a curve the na«
tuie of which is supposed to be defined by what is given con-
cerning the variable quantity. A curve line either returns into
itself, or may be continued without end ; and theref(»re there
are ^ways two branches of the curve that proceed from any
point that is assignable in it. The ordinate from a point of the
eorve is a maximum, or minimum, when it is greater or less than
the<»dinates which may be drawn from the parts of either branch •
ofthecurveadjoiningtotbat point. (JRfg. 81 and 82.) When tha
curve is continued immediately from that point on both sides of
the ordinate, we vhall call the ordinate a maximum, or minimum,
of ihefirst kind, but of thesecond kindwhen the curve is reflected
from that ordinate (fig. S3), and both the branches of the curve
are on the same sideof it. Sometimes four or more branches of
acurveproceecrfromthe8amepoint;but the ordinate in that case
isonly to be compared with the ordinates from these twobranches'
that are to be considered, according to the nature of the curve,'
as the immediate continuation of each other.

241. The writers on this subject do not always agree as to
the extent of the problems they comprehend under this class;
Some * have proposed to comprehend under the maxima and
minima the greatest or least ordinates that can be drawn firt»m
the part of the curve that is convex towards the base, or fronr
that which is concave towards the base, though the ordinates
from the adjoining parts of the succeeding branch of the curve
may be greater or less than thftt ordinate. Others, who ex-
clude thut case^ seem also to exclude the greatest or least ordi-
nate when it passes through a point where the curvature is not
continued, that is, a point of Reflexion, or Cuspis. Others com-
prehend the ordinate from a cuqpis when the twb branches of
the curve that proceed from it are on different sides of the or-
dinate, but exclude such an ordinate when the two branches of

* I>McaTt«s*s Lettetty torn. 9, bt 60.

O 4 Ae

216 Of the greatest and hast OrHnates. Book L

the e^r^e dwt proceed fmmthe cttspbareonthesameakkctf the
ordio^te, because th^^se branches (or atleast the acyoiningparts
oi each) are over the same base. But it seems to be more con^
sisient^ to include all ordinates that are greater^ arless^ than
those from the adjoiniug parts of either branch of the curve
(Jig* S3); the rather^ that in the latter case one of the branches
of the cunre^ after beingreflected from tbeordinate^ oftenietama
to it again^ and after cutting it^ proceeds on the other side^ so
that one or more ordinates may correspond to any asB^;naUe
base« However^ since it has b^n more usual to exchide this
case> we have distinguished the greatest and least ordinates
into two kinds in the preceding artide^ to prevent mistakes*

^^- (Jig* 31). When an arcn of a curve has its concavi^
turned one way^and there is apointin this arch wherethe tangent
becomes parallel to the base, the greatest or least ordinate
passes through that point. It isgreater orlessthan those&omthe
adjoining parts of the arch on either side, according as the ardi
is concave or convex towards the base. Supposing the base to
increase, theinrdinate in the former case first increases and then
decreasea,and in the latter case, theordinate first decreases and
then increases. In both cases, the motion with which the base
flows (or its fluxion) being given, the motion with which the
ordinate flows (or its fluxion) first decreases and then increases,
by the 7 th lemma : and this motion, or the fluxion of the or-
dinate vanishesi when the tangent becomes parallel to the
base. For^ £1 (fig. 47 and 50) being supposed to measure the
fluxion of the base, IT which measures the fluxion of the ordi-
nate (b}^ prop. 14) vanishes when the tangent £T becomes pa^
rallel to the base, and coincides vrith EI. Therefore the great-
est and least ordinates are discovered in such cases (which are
thosethatmost commonly occur in the resolution of proMems),
by enquiring when tbe fluxion of the curve becomes equal to
the fluxion of the base, or when the fluxion of the ordinate va-
nishes, the fluxion of the base being given. In this ca9e,how«*
ever, the curve must be continued on both sides of the ordinate:
for, if the curve be rejected from the ordinate D£, then the
point E is a cuspis, and tfaie ordinate D£ is neither a maximum
nor minimum when the aTche8CE,£H have their convexity to-
wards

Chap. IX. Of the greateii and Ua$i Ordinates, 617

wards each other (/tg. 84, fig. 85); and it is a maximufh or
minimum of the second kind when those branches are on the
same aide of the tangent at E, the convexity of one branch
being towards the concavity of the other.

£43. The arch being supposed to have its concavity turned
one way^ and the tangent at E {fig. 81> n. 1) being supposed
parallel to the base^ if the arch meet the base we may conclude
that D£ is a maximum (fig, 81^ n. 2); but if the arch be of that
kind which maybe continued above the base without end^ DE is
a ndnimum. The greatest and least ordinates are distinguished
from each 6ther more generally^ by comparing them with the
ordinates from the adjoining parts of the curve ; or by supposing
the baseAPto increase^ and observing whether the fluxion of the
ordinate PM is positive before the point P comes to D, and be-
comes negative after P passes D, or is first negative and then be-
comes positive^ the ordinate being itself considered as positive : '
it is a maximum in the former^ and a minimum in the latter case*
244. In the application of this rule for finding the greatest
and least ordinates^ it must be observed^ that the ratio of the
fluxion of the ordinate to the fluxion of the base maybe some-
times represented by the ratio of two quantities, which at cer-
tain terms may both vanish together : but it does not follow,
that in such cases the tangent becomes parallel to the base, or
that the ordinate is a maximum or minimum ; for the ratio of
those fluxions may be represented in such cases by that of other
finite and assignable quantities. When two or more branches
of the curve are over the same part of the base^ the ordinate must
arise to two or more dimensions in the equation of the curve.
'The fluxion of the base being given, the fluxion of the ordinate
isatleast twofold ; and when the ordinate passes through a point
where two or more branches of the curve intersect each other,
we are to expect, that though the base and ordinate be both
given, the fluxion of the ordinate will arise to two or more di-
mensions in the general equation by which it is to be determin-
ed. But if the general equation of the curve can be resolved
into* these particular equations which belong to the difierent
branches of the curve from which the general equation is com-
pounded, then the fluxion of each ordinate may be determined

from

218 Of the greatest and leaU Ordinates. Book I.

from the fluxion oi the equation of each branch. But we maj
have occasion to illustrate this further afterwards.

£45. (JPig- 82). When the fluxion of the curve coincides with
the fluxion of the ordinate^the tangent becomes perpendicular to
the baseband coincideswith the ordinate. In thiscase^if £C^£H
the two branches of the curve form a cuspis at £, being on dif-
ferent sides of the ordinate DE^ then is DE a maximum or nU-
mnwn according as those branches have their convexity or con-
cavity towards the base. The maxima and minima of this sort
are discovered, by enquiring when the fluxion of the curve be-
copies equal to the fluxion of the ordinate, or (which is the
fame thing) when the fluxion of the base vanishes, the fluxion
of the ordinate being given. In this case, also, it is necessary
that the curve be continued immediately from E on both sides
of the ordinate DE ; for if the two branches EC, EH be on
the same side of DE, the ordinate is either no maximum nor
minimum^ or is one of the second kind.

246. These are the two rules that are commonly given for
determining the maxima and minima of the firstkind. But there
IS still anotherlimitation besides those already mentioned, with*
out whiph these rules may lead us into error. For we are not
always to conclude that the ordinate DE isa maximum or mini-
mum, either when the fluxion of the ordinate vanishes, the flux*
ion of the base being given ; or when the fluxion of the base
vanishes, the fluxion of the ordinate being given, though the
curve be continued immediately on both sides of the ordinate:
in the former case the tangent at E is parallel, in the latter, per-
pendicularto the base; but E C^g.9l) may be a point of contrary
flexion, so that the ordinates on one side of DE may be greater
than DE, and those on the other side less than it; and there may
be no maximum nor minimum perhaps from the wfiole curve.
In any curve that has a point of contrary flexion, the ordinate
discovered by those rules is not a maximum or minimum, if the
base be parallel or perpendicular to the tangent at that point.
The rules which are given for finding the pointsof contrary flex-
ure are liable to exceptionsof the same nature for a similar reason.
In order to set these rules and the necessary exceptions in a clear
hght, it will be of use to premise the following proposition.

PROP.

Chap. IX. Gfthc greatest and least Ordinata.. « 19

PROP. XX.

S47. Let the ordinate PE (fig. 86) 9/ the curve AEH tneet the
curve Teh in e, and the rectangle £6 contained by DE and 4.
given right line DG be always equal to the area ADcF;
then, the rectangle JieLG being completed, £T the tangent
of the curve AEH at £ shall be parallel to the diagonal DL.
And, conversely, £K being equal and parallel to DGj if
KT parallel to DE meet the tangent ET in T, and, De
being taken always equal to KT> if the curve HEC meet
the base in A, the area ADcF shall be equal to the rect--
angle EG.

The fluxion of the base AD being represented by the given
Tight line DG or EK, the fluxion of the ordinate DS is repre-
sented by KT^ by prop. 14^ and the fluxion of the area AD^F
by the rectangle eG^ by prop. 3. The rectangle EG is always
equal to the area ADe'P, by the supposition ; and the fluxion of
EJGr is equal to the fluxion of the area AD^F^ by art. 18 ; that
is^ the rectangle contained by KT and DG is equal to eG,
and KT is equal to De or GL. Therefore the tangent ET is-
parallel to DL.

. £48. And, conversely^ the right line DG or EK being ^ven,
if upon the ordinate DE a right line De be taken always equal
to KTj the rectangle eG shall be always equal to the rectangle
<lontained by KT and DG ; that is^ the fluxion of the area
ADeY shall be always equal to the fluxion of the rectangle EG;
and, by theor. 4, the fluents generated in the same time being
equal, it follows, that since the curve CEH is supposed to
pass through A> and the point E sets out from A when e sets
out from F, and the right line D^ frdto AF, the area AD^F
must be equal to the rectangle EG. The analogy there is
betwixt the inverse method of tangents and the quadrature of
curvilineal figures appears from this proposition.

. 249. It may be of use for illustrating this doctrine, to demon**
etrate in the following manner (which is independait of the

method

<dO Of the greatat and UaU Ordmata^ Book I.

method of flaxions)^ that if the rectangle E6 be always equal
to the area ADeF, then ET parallel to DL shall be the tan*-
gent of the arch EH at E. For^ let any ordinate PM meet
the arches EH^ eh, the right lines EK^ eL parallel to the
base^ and £T parallel to DL^ in the points M,Vi, V, S and
R respectively. Then, since the rectangle contained by VR
and D6 is to the rectangle contained by KT and DG as VR
b to KT, or DP to DG, or as the rectangle «P is to c6 ;
und the rect^gle contained by KT and D6 is eqnal to cG ;
it follows, that the rectangle contained by VR and DG is
equal to the rectangle eP. But by the suppo^on the *rect-
angle contained by PM and DG is equal to the area APNF ;
and the rectangle contained by DE and DG is equal to the
area ADeF : therefore the rectangle contained by VM and
DG is equal to the area DPNe; and VM is to VR as the area
DPNe IB to the rectangle ^P. But when the ordinates from
the arch eh increase while the base increases^ the area DPNc
always exceeds the rectangle ^P. Therefore, in this case, VM
is always greater than VR, and the arch EH is all above the
right line ET, the point E only excepted. Nor can any r^t
fine ht drawn through E within the angle of contact HET.
For,sinoe KH is to KT (or 6L) as the area DGhe is to the
rectangle eQ, and, therefore, in a less ratio than Qh is to 61^
it follows, that KH is less than GA. Therefore, Q being any
point betwixt T and H, if DO be taken equal to KQ, it shall
be less than Gh, and a right line through O parallel to the
base shall meet the arch eh in some point betwixt e and A. Let
this point be N, and let the ordinate PN meet the arch EH bbA
the right lines EQ, EK, eL in the points M, Z, K and S. Then,
fince the rectangle contained by KQ and DG is to the rectangle
contained by VZ and DG as KQ is to VZ, or DG to DP, or
as the rectangle OG is to OP^ it follows, that the rectangle
contained by VZ and DG is equal to OP, and that VZ is to
VM as the rectangle OP is to the area DPNc. Therefore VZ
is greater than VM, and the right line EQ does not pass
through the angle of contact HET, but cuts the arch EH in
some point betwixt M and H. From which it follows, by
Urt. 181^ that ET is the tangent of the arch EH ai E*

240. In

^aq^pC. ,Qf the greaiut and least (MSn^ta. . t&l

fi50* In the swie maimer^ if any ordinate pn meet the arches
ee, CE and the right lines "Bt, £^, el (which are the right lines
T£> KE^ Le> co&tinned bejondEand e) in fi, m, r, v, wad/, vm
shall be to tr as the area Dpne is to the rectangle ep ; and,
therefore, when the ordinates fix>m cfi increase while the basa
increases, the arch EC is all above Ei, the point £ excepted.
In the same maimer^ kC is greater than Be (the rectangle con-*
tained by kC and DG being equal to the area DBce which ex*
<;eed8 the rectangle contained by Be and DG) ; and kt being
^ual to KT or De, if g be any point betwixt C and t, and Do
he taken equal to kq, a parallel to the base through o shall meet
the arch ce in some point betwixt c and e. Let that point be
n, and let the ordinate pn meet Ejr in z; then, since kq (or Do)
is to «8? as DB is to Dp, or as the rectangle oB is to op, it fol-
lows, that the rectangle contained by vz and DG is equal to
the rectangle op, and that vz id to twias op is to the aieaDpne*
Therefore vz is less than vm; and, since AC is less than kQ, the
right line ^q does not pass through the angle of contact CEf,
hut intersects the arch EH in some point betwixt m and C.
Therefore £^, which is the right line ET continued, is the tan^
gent of the arch EC which is the contTnuation of the arch HE^
Becaose the tangent tT is betwixt the curve CEH and the
base in this case, it appears, that when the ordinates from the
arch ceh increase (the base AP being supposed to increase),
the arch CEH is convex towards the base.

iS6 !• When the ordinates from the arch ceh (fig. 87) deca^ase
whilethebaseinGrease8,therectangIeePexceedstheareaDPNtf,
and the rectangle ep is less than the areaDpn^ . Therefore VR is
greater than VM^ and vr less than vm ; so that the whole arch
HEC is below the right Kne T£^, the point £ excepted : and
it is shown in the same manner as in the preceding case, that
no right line can be drawn through the angles of contact HET,
CE^. Therefore, in this case, the right line T^ is the tangent
at E, and the arch CEH is concave towards the base* In both
eases the curve CEH passes through A, because when AP va«
nishes, the rectangle contained by PM and DG (which is sup-
posed to be always equal to the area APNF) vanishes ; and if
A6 be taken towards G equal to DG^ and the rectangle AbfJi
he completed, the diagonal A/* shall be the tangent at A.

«5e. Cor.

tS2 Of ike greatest and least OrHhaiei. Bopk t

S5£. Cor. I. Let Zez(Jig. 86) the tangent of the arch ceh^ii
meet GA and Be in Z and z, and let the ordinates fiom Be to
GA increase continuany ; theiij if the arch ceh be conrex to*
ward? the base^ TH which subtends the angle of contact HET
shall be less than one half of Lh, but greater than one half of
LZ. For the rectangle contained by TH and D6 is equal to
the area eLh which is less than the triangle eLh, or one half of
the rectangle contained by DG and LA^ but is greater than the
triangle elZ, or one half of the rectangle contained by DG and
LZ. Therefore TH is less than one half of Lh, but greater
than one half of LZ. In the same manner it ^pears^ that tC
is less than one half of Iz, but greater than one half of ic,

£53, Car. H. When the arch ceh (Jig. 87) is concave towards
the base^ TH is less than one half of LZ^ but greater than one
half of L4. For^ in this case^ the area eU is less than the tri-
angje eUZ, but greater than the triangle eLh, In like manner
'iC is in this case less than one half of Ic, but greater than one
half of &.

' 254. Cor, IIL When ceh is a right line^ and coincides with
xeZj CEH is an arch of a parabola that has its axis perpendi-
cular to the base AG. In this case TH and ^C are each equal
to one half of LZ which measures the fluxion of De (by prop.
14)> or the second fluxion of D£^ the fluxion of the base being
represented by DG. While the base AD acquires the augment
DG^ the ordinate DE. acquires the augment KH equal to the
sum of KT and TH ; and in this case the first fluxion of the or^
dinate is represented by KT^ and its second fluxion by £TH> or.
the sum of TH and ^C. But when ceh is conyex towards the
base^ and the base being supposed to flow uniformly^ its flux*
ion is represented by DG^ the right line TH is greater than
one half of LZ which measures the second fluxion of the or-
dinate DE^ but less than one half of L& which measures the in*
crease of the fluxion of the ordinate that is generated in the
same time in which the base acquires the augment DG. When
the arch ceh is concave towards the base ^ TH (which subtends
the angle of contact HET) is less than one half of the right
)izie 'that mea3ures the second fluxiim of DE^ but greater than
<nae half of the right line that measures the increase of the

fluxion

I

Chap. IX. Of the grtatest and least Ordinate^. S23

fluxion of DE. In the first case^ when ceh is a right line^ the
motion with which DE flows is uniformly accelerated. When
ceh is convex towards the base^ the acceleration of that motion
increases; and when ceh is concave towards the base^ its ac*
celeration decreases continually.

255. Cor. IV. When the curve Ach (fig. 88) is a parabola that
has its axis perpendicular to the base AG, KH the increment of
DE may be distinguished into three parts^ KT^ TQ^ and QH^ so
that the rectangle contained by those parts and DG may be re-
spectively equal to the rectangle cG, the triangle cUL^ and the
area f ZA. The part KT is equal to J}t, and measures the first
fluxion of DE; the part TQ is equal to one half of IIL, which
measures the second fluxion of DE(by cor. 3)^ and the part QH
is equal to one third part of U (that measures one half of the
fluxion of LZ)^ and therefore measures one sixth part of the
third fluxion of DE. For it follows from what was shown in the
Introduction (page 27), after ArchimedeSy that the area cU is
one third part of the rectangle contained by 7Lh and.DG : and
it may be easily deduced from the 8th proposition ; for let PM
meet the arch eh in N^ and its tangent eZ in u; and tiN shall
be to ZA as the square of DP is to the square of DG. Therefore,
the point D and the right lines D6^ U being given, but sup-
posing DP to flow, and DG, DP, X, and Y to be in continued
proportion, it will follow, from the eighth proposition, that
one third part of the fluxion of Y shall be to the fluxion of
DP as Y i^ to DP, or as the square of DP is to the square of
DO, and therefore as uH is to ZA ; so that the rectangle con-
tained by uN and the right line which measures the fluxion of
DP b equal to one third part of the rectangle contained by
U and theright line which measures the fluWion of Y. There-
fore the area ^uN is equal to one third part of the rectangle
contained by ZA and Y, or (because U is to uN as DP is to
Y) of the rectangle contained by uN and DP, and the area
eU is equal to pne third part of the rectangle contained by
ZA and D6. In the same manner it is shown, that the area
£RM is equal to one fourth part of the rectangle contained
by RM and DP ; and the continuation of these theorems is
obvioiu from the same eighth proposition. From which it

- " follows.

1

I

224 Of the greafat and least Ordinata. Book I.

foUowSj that when the fluxions of all orders of the ordinate DS
increase^ we approximate continually to the ralue of KH> the
increment of the ordinate that is generated in the same time
the base acquires the augment DG^ by adding continually to-
gether the right line that measures the first fluxion of D£ whik
PG measures the fluxion of the bascj \ of that which measures
the second fluxion of- the ordinate^ \ of that which measures
its thu'd fluxion^ 4^ of that which measures its fourth fluxion,
and so on^ the denominators of those fractions being the pro*
ducts of the numbers 1^ 2> Sy 4« 5, i^c. in their natural order.
But when any fluxion decreases^ the succeeding fluxion is to
be cousidcxed as negative^ aoid the iraction which involves it is
to be subducted. These corollaries illustrate what was shown
of second and third fluxions nea^^^fac'Ad of the first and
fourth chapters.
256. Let any point £ be given in the curve CEH (fig.99)f let
KEk be a right Une paraJ^ tothe base ADj PM an ordinate
meeting EK in V and the curve cM in N^and the rectanglecon*
tainedbyVKf and the given rightlineDG be always equal to the
area DPN^. Suppose the points P and M to move from D and
E ; and, when AP increases^ and PN is above the base AP,
let VM be taken upon PV produced beyond V: then, when
AP decreases^ if PN be still above the base, or when AP in-
creases if PN be below the base^ VM is to be taken upon VP
from V towards P; but if AP decreaie, and PN be below
the base, VM is to be taken upon PV produced beyond V.

£57. When the point e in the cwcxe FcA (fig. QO) fells on the
base, and coincides with D,DLcoincides with DG,the tangent
/ET coincides with A£K, and becomes parallel to the base. In
this case, if the curve FcA after meeting the base AD, be con-
tinued on the other side of AD, and on the other side of the
perpendicular D£, then is D£ a maximum ; for, by the last
article, P being taken on either side of D, VM is to be taken
from V towards P. When PM meets the curve FcA below
the base in N and the curve CEH above it in M, the rectangle
contiiined by VM and DG being equal to the area DPN, it
follows, that the rectangle contained by PM and DG is equal
to the excess of the area ADF above the area PPN, ai^d that

PM

Chap. IX* €fihc greatest and least Ordinatesi £25

PM Taniahes when those areas become equal. If the curve FBN
return towards the base, and after cutting it again in d be 6on«
tinued on the opposite side of the base and of the perpendicu-
lar de, then de the ordinate of the curve AEM at d is a tnaxi^
mum or minimum, according as it meets the curve above or be-
low the base. In the same manner, if the curve NF continued
beyond F meet the base in a, and proceed from a below the
base on the other side of the perpendicular at a, the curve MBA
shall be continued from A bdow the base, and its ordinate at a
shall be a maximum.

258. By the 256th article, the form of the arch ceh (Jig. 89)
being given, we discover the form of the arch CEH. Suppose
now. the form of the arch CEH to be known, and that of ceh to
be required. Let £[ 6e ftny point in the arch CEH, and suppose
that De is taken above the base when KT is upon 6K ptodaced
beyond K. Then, if the tangent ISt meet tiK in T betwixt G
and K, or 6A produced beyond k in /, the ot'ditiate De is to be
taken upon ED produced beyond D below the base; but when
ET meets BA in < betwixt B and k, De is to be taken above the
base.

259. From this the converse of the 257 th article is manifest
(Jig. 90). That when'the ordinate DC is a maximum or minimum
of the first kind, and the tangent at E is paraDel to the base, the
arches ec, eh of the curve Ceh must be continued from e (which
coincides with D in this case) on different sides of the base AD
and of the perpendicular at D. For, while the point M describes
the arches CE, EH, the ordinate PN of the curve ceh must be
taken on different sides of the base.

260. But when thecurveFND(^.91);aflermeetingthebase
in D, is continued on the opposite side of DE, but on the same
side of the base as before^ then the ordinate DE is not a maxi"
mum or minimum,thovLgh PN which measures the fluxion of the
ordinate vanishes when P comes to D. Tor, while N describes the
arches cD and Dh, the right line VM is to be taken on different
sides of kK, which is parallel to the base. Therefore DE is
not a maximum or minimum, but E is a point of contrary flexnrej
whether the arch cDk touch the base in D, or form a cu^is
there. It is manifcHy conversely/ from art. 258> that, when IL

VOL I. P ii

^

226 Of the greatest and least drdinates, feook I.

is a point of contrary flexure, and the tangent at E is pardlel to
the base, the arch ch, after meetiilg ttoebasein t), is continned
on the same side of the base; because the ordinates from BC to
GH increase or decrease continually : and, MI bei ng equal ant!
parallel to DG, if IT parallel to DE meet the tangent at M in
t, the right line It is on the same side of MI whHe M describes
the arches CE and EH.

PROP. XXL

126 1. Tkejluxion of the base being given, and the curve being
continued on both sides of the ordinate, when the first

fluxion of the ordinate and its fluxions of any number of
subsequent successive orders vanish, the ordinate is a maxi-
mum or ininimum, or passes through a point of contrary

flexion, according as that number is even or odd.

(JPig. go and 91). The curve being continued on both sides
of the ordinate, and the fluxion of the base being given, let the
first fluxion of the ordinate DE vanish ; then, if the number of
its subsequent successive fluxions that vanish be 0, % 4, or any
even number, the ordinate DE shall be a maximumor minimum;
hut if that number he 1, 3, 5, or any odd number, E shall be a
point of contrary flexure. First, let the curve ch cut the base in
D in any assignable angle, and be continued fromD on opposite
sides of the base and ordinate. In this case DE is a maxt*
mum or minimum, by art, 257^ and the first fluxion of DE var
nishes, but the second fluxion does not vanish. In the next
place, let the arch CH be substituted in place of ch, and a new
curve be derived from CH in the same manner that CH was de-
rived from ch: then DE shall meet this third curve in a point
of contrary flexure, by art. 26O, and the first and secopd flux-
ions of the ordinate of this curve vanish. If this third curve
he substituted for ch, and a fourth be derived from it in the same
manner, the ordinate of this fourth curve at D shall be a max^
\^imum or minimum, and its iHrst^ second, and third fluxions, va-
nish.

TlateXftziaKTaJl.

%

I «

^

. Chap. IX; Md tlu FoinU of contrary Flexion, 227

nish. This series of curves bdng continued, it is manifest that
D£ shall be a tnaximiim or minimum, and pass through a point
of Gontraiy flexure^ alternately ; the number of fluxions of suc-
cessive orders (not including the first fluxion) that vanish being
alternately an even and an odd number.

262- When De {fig. 92) is an asymptote of the curve FcN,
there are always two branches of the curve^ as cH, AN, which
approach to the asymptote when they are produced*. If the area
bounded by the base AD, the ordinate AF, the asymptote Dc,
^md the curve FcN, is always less than a certain finite space to
which that area continually approaches while the curve and its
asymptote are produced, so that theirdiflerencemaybecome less
than any assignable space; and if the two branches cN,AN pro-
ceed along the asymptote on different sides of it and in differ-
ent directions, then the ordinate DE is 9^maximium or minimum,
«uch as was described in art. 245 (Jig. 93). When thosebranches
approach to the asymptote on difierent sides of it, but proceed
along it with the same direction, E is a point of contrary flex-
ure, and DE is not a maximtm or minimum. In bbth cases,
the tangent at E is perpendicular to the base, and the fluxion of
the base vaniriies, when compared >vltb the fluxion of the ord|^
nate ; but it is only in the former case, when the fluxion of the
£>rdinate is positive on one side of I>E and negative on the
other side of it, that DE is a maximutn or tnimmmi' We have
often observed, that the fluxion of a quantity is considered
as positive when the quantity increases, but as negative wheti it
decreases: and these fluxions are represented by right lines
-that stand on opposite sides of the base, the contrary positions
of which answer to the opposite afiections of the fluxions. It is
common to the maxima and minima described in the 242d and
245th -arlides, that, the fluxion of the base AP being positive,
the fluxion of the ordinate PM is positive on one. side of D£
>and negative on the other: but in the former it vanishes when
PM coincides with DE; and in the latter it is said to hecx^me
infinitely great inthat case, as the ordinate of a curve is said to
4>ecome infinitely great when it coincides with an asymptote.
And it is- said in general^ that a quantity thatis podtive become^
negativeyeilherby^creasiAg and passing, by nothing, or by4n^

P 2 creasing

€28 Of the Points of contrary Fktion. Book L

creasing and passing by infinity : ns(J^. 94), if w^ suppose aright
line FQ revolving about a given point F always to intersect the
right line Oo in P^ and FA to be perpendicular to Oo in A^ the
right line AP> being first considered as positive, becomes nega-
tive by decreasing and passing by nothing, when FQ being sup-
posed to move from FO towards Fo, the intersection P passes
through A to the other side of A ; but AP is said to become ne-
gative by increasing and passing by infinity, when FQ moves in
fL contrary direction, and, after becoming parallel to Oo, meets
it again on the other side of A. It 16 ^ual to explain analogies
.betwixt figures in the common doctrine of curve lines in this man-
ner, and it is often of use in transferring readily the properties of
one curve to another that is under the sameg^us. Thus authors
explain how theellipse is transformed into aparabola, and thence
jntoanhyperbola.But,IestconceptioDsorexpressionsofthiskind
should be excepted againstj webave endeavoured to avoid them.

PROP. XXIL

263. The ordinate meets the curve in a point of contrary
fkxure when its fluxion is a maximum or minimum, the
fluxion of the base being given, and the curve beihg continued
on both sides of the ordinate.

(JFTg.95). Resuming the construction of the 249th and 25l5t
articles, it foUowsfrom what was demonstrated there, that, when
the or^inates of the arch ce increase from Be to De, the arch CE
is convex towards the base, and that when theordinates from Be
to GA decrease, the arch EH is concave towards the base; that is,
when De is a maximum, and the arches ce, eA are on difierent
sides of Df , the point £ is a point of contrary flexure. But De
represents the fluxion of the ordinate DE,tbe fluxion of the base
being represented by the given right Une DG. llierefore, when
the fluxion of the ordinate is a maximum, and the curre k cod
tinned from the ordinate on botli sides, it meets tbe carte in a
point of contrary flexure (ftg, 96). In like maana:, when the
ordaiates from the arch ce decrease, and those from eh increase

(th«t

Chap. IX. Of the Points of contrary Flexion* Vtp

(that 18^ when De is a fninimum), the arch CE is concave and
the arch EH is convex towards the base^ by what was shown in
the 249^ and S5l8t articles. Tlicrefore, when the fluxion of
the ordinate is a mnimum, the fluxion of the base being given^
and the curve being continued from the ordinate on both sidesy
the ordinate meets the curve in a point of contrary flexure. : ^
264. {Fig.gs) The proposition appears also from the conveite of
the 7th lemma, art. 1 84. For^ if we suppose the fluxion of the
ordinate DE to be afnaximumj\he fluxion of the ordinate PM
must increase while M describes C£^ and decrease while M de-
scribesEH. Therefore^ by the converseof the7th lemma,if PM
increases while M describes CEH^ the arch CE must be convex
and EH concave towards the base ; "so that E must be a point
of contrary flexure. If PM (fig. 96) decrease while M de-
scribes CEH, the arch CE most be concave and the arch EH
convex towards the base, and E a point of contrary flexure.
In the same manner, when the fluxion of DE is a minimum, it
appears that E is a point of the same kind. We do not com-
prehend under the maxima or minima quantities that vanish, or
such as are supposed to exceed all assignable magnitude.

265. Cor. I. As there are various kinds of maxima and mi*
mma, so there are various kinds of points of contrary flexure.
As in the mostcommon cases, the ordinate isamaximfm ormt-^
lumifm when its fluxion vanishes, the fluxion of the base being
given ; so, when the second fluxion of the ordinate vanishes,
the ordinatemost commonly passes through apointof contraiy
flexure. But this is not universally true, though .the curve be
continued "^n both sides of that ordinate. For when the tan-
gent of the curve uk at e becomes parallel to the base, the se-
cond fluxion of DE, or the first fluxion of D^, vanishes, the
fluxion of the base being given; but in this case, t (fig. 97) may
l>e a point of contraiy flexare (art. 260), and,*De (which mea^
sures the fluxion of D£) not being a maximum or mininmm,
the point E is not a point of contrary flexure, but the whole
arch CEH has its concavi^ turned the same way.

9S6. Cor. II. (fig. 95, 96, and 97) In general^ whether the
first fluxion of the ordinate vanish or not, the point £ is a point
of contraiy flexure when the number of the subsequent suc-

P 3 cessive

aSO Of the Points of contrary Fleiiou Book L

oessi^e orders of fluxions of D£ that vanish is an odd niunba*,
if the curve be continued from the ordinate on both sides ; but
£ is not a point of contrary flexure^ and the whole arch CEH
has its concavity tamed the same way when that number 10
even ; for in the former case Dt is a maximum or minimum,
and in the latter case e is a point of contrary flexure, by art. 26 1 •

£67. When De or the fluxion of the ordinate D£ is such a
maximum or miniinum as was described in the ^5th and 962d
articles^ £ is a point of contrary flexure^ and^ the tangent at e
being perpendicular to the base^ the right line that measures the
fluxion of the base vanishes in this case when that which mea*
sures the fluxion of De> or the second fluxion 6f DE, is given.
But it does not follow^ conversely^ that when this happens, E
is always a point of contrary flexure, although the curve be con-
tinued on both sides of the ordinate D£. For if e itself be a
point of conti^ flexure inthecurve FN 0Sg.98)> then^thou^
the tangent at e be perpendiccdar to the base, De is not a nMurt-
mium or minimum, and E in due curve CEH is not a point of
contrary flexure ; but the whole arch CH has its convexity or
concavity towards the base, according as the ordkiates of the
arch ch continually increase or decrease from Bc^ to Qh.

%66. Hitherto we have supposed the two arches C£ and EH
to be on diflerent sides of DE, When these arches ase on the

samesideofDE^andhaveatangentatEdiflTerentfromDBjthen
Eis apoint of reflexion, or cuspis ifig.QQ, and 101). Thecele*
brated author of the Analyst dcs ifffiniment petiis distinguishes
diose points into two kinds ; the point E is a cuspis of the^rst
kind when the arches EC, EH have their convexity towards
each other, bnt of the second kind, when the ccmvexity of the
on'e is towards the concavity of the other.

269. Let the arch ch meet the base in D, the points D and e
being suppofiied td coincide ; and let the arches ce, dk be on dif-
ferent sides of the base, but on the same side of DE. Then, whe-
tiier the arches ce, eh be perpendicular to the base, and ifonnoaue
cpntinued arch ch, or touch the base, and form a cuspis of the
flrst kind at e, the point £ is a cuspis of the first kiiid.^Fbr, when
P^ (J*g' 99j n. I and 2) is above the base, VM is to be taken upon
VP fx-om V towards P, and 'when PN is below the base, VM is to

be

J

^hop. .IX. and R^xion. g^

be takei^ upon PY produced beyond V^ by arL £^. ^d>.oo&r
versdy, vfheu £ is a cu9pis of ^he first kind^ if the tai^Qt aii
)S be parallel to the base^ when the base AP increaaes^ the pxr
dinate fiom oq^ of the arches C£, HE iacreases^ and the on)ir
nate irom the other decreases. The fluxion of one of the o^v.
dinates is positive, and the Ihixionof the other is neg^ye^ 19
that the right Unes which represent these fluxionsmust be takea
on opposite side^ of the base ; and ce, eh must either form a co^-
finned arch ch, 9t a cu^is of the fyc^i kind at e. }n bot|i casef
the first fluxion of D£ vanishes; and in the former^ the fluxio;^
of J)<, or the second fluxion of D£, being given^ the fluxion
of the base van^bey ; in the latter case^ the second fluxion 0^
P£ yanishes as well as the first fluxion^ and, the third fluxion
of I)£ being compared with the fluxion of the base^ ijtt^ on$
yimishes when the other is given. Jn general, w|ien £ls a cus*
pis of the first kind^ and the tangent at £ is parallel to the base;^
the fttt^ns of D£ of any number of successive orders may va^
nishi the fluxion of the base bejjig given ; but the fluxion of thf
meifi order to those that vanish cannot be to the fluxion of th^
base in any assignable ratio, and is said (according to theusus^
langw^ 90 t}iis subject) to become infinitely great, in the
same sense as the ordinate of a curVjC is commonly said to be-
Gom» infinite whei) i^ is ^i;(pppsed to coincide with ajna^jmptpte*

PROP. xxin.

tf70. The ardvu cty th (fig. 100) being on the mme si^ tfpe,
the point £ ic a outpis of the first kind when cehis a contimioji
arch, but of the tecond kind when eisa cuspis cfthe fint kind,
wd the tangent at e is oblique to De, or when e is itself a
euspisofihe second kindi

Case 1. When ceiis a continued arch, and touches the ordi-
nate De, VM is tol^ taken upon VP from V towards P both
when N describes the'^l^art ce and the part eh, by art. 256. The
right line Et parallel to DL^is the tangent of both the branches

P/4 CE

1

«3ft Of the Points of contrary Hexian Book!

CJE Md EH, by prop. 20- Therefore the point E is a cospis.
lle'areas D^^B, DifB, Df AB are respectively equal to therect-
angles contained by the right lines KQ Kt, KH, and the given
right line D6. Therefore KC is less than Kt, and K^ less than
KH ; so that the tangent E^ must pass betwixt the arches EC,
EH ; and, these arches being therefore convex towards each
other, E is a cuspis of the first kind.

* 271. Case^. lete(Jig. 101)beacuspisofthefirstkind,andifthe
tangent at e be not parallel to the base, thepoint Eshallbea cus*
pb of the second kind. For let the perpendicular at P meet the
kfches ce,eh, CE, EH in the points n, N,m, and M, the tangent
& in R, and EK, el parallel to the base in V and S. Then, by
art. 256, the right lines VM, Vm are to be taken from V to-
wards P when N describes ce, or n describes eh. The rectangles
contained by VM,Vm, and VR, and by the givea right line DG,
arc respectively equal to the areas Df NP, DenP, and DcSP ; and
Ihe areas DeNP, D^P are either both less or both greater thaii
DifSP. Therefbre VM, Vm are both less or both greater than
¥R, the arches EM, "Em are on the same side of the tangent
'Et, and, consequently, E is a cuspis of the second kind.

272. Case S. When e is a cuspis of the second kind,it appears
in the same manner that the arches EM, Em are on the same
side of the tangent E^, and therefore E is a cuspis of the same
kind.

273. Cor. I. In the first caae, the point E is a cuspis of the
first kind,'and the right line which measures the flu^^ion of the
base vanishes when that which measures the second fluxicoi of
the ordinate is given. The point E is also a cuspis of the first
kind when the right ime which measures the second fluxion
of the ordinate D£ vanishes, the fluxion of. the base being
given, if the two arches ee, eh are on thesame side of DE, and
are convex towards each other. But E is not always a cuspis
of this kind when the right line which measuresllie second flux-
ion of the ordinate vanishes, that which measures the fluxion
of the base being given, or when the latter vanishes, the former
being given, though the arches CE, EH be on the same side of
DE. For in this case e and £ may be each a cuspis of the stn
cond kind.

274. Cor.

Chap. iX* mnd Reflexion. 833

^4. Cor. 11. In the second case^ E isa caspis of tfie! second
Icind^ and the right Hne which measures the third flnxion of the
ordinate D£ vanishes when that which measures the fluxion of
the hase is given, or the latter vanishes^ the former being given.

275. Cor. ni. C^g.lOOand 101). WhentherightlinevHliidi
measures thesecondfluxion of theordinate DEisin an assignable
ratio to that which measures the fluxion of the base, and the
arches CE, EH, are on the same side of the ordinate, then the
point E is always a cuspis of the second kind, wbetherthethiid
fluxion of DE be assignable or not; for in that case the tangent
of the arch ce at e is oblique to Df , the point e is a cuspis either
of the first or second kind ; and therefore E is a cuspis of iht
second kind, by art. 271 and 272. When the ratio of the right
%ne which measures the second fluxion of the ordinate I>£%6
that which measures the fluxion of the base is not assignaUe
<the arches CE, EH being still on the same side of DE), if the
ratio of the right line which measures the third flnxion of DS
to fhal which measures the fluxion of the base be assignable,
£ is still a cuspis of the second kind; but when thislatterratib
also is not assignable, E may be a cuspis of either kind. *

276. The ordinate DE is a maximum or mimmum of the se*^
cond kind in all cases when E is a cuspis of either kind, that
only excepted wherein E is a cuspis of the first kind, and the
tangent at E is tt the same time parallel to the base. Andwe
may conclude DE to be a maximum or minimum of this kind,
if the curve is not continued on both sides of the ordinate, not
only when the first fluxion of the ordinate is to the fluxion of
the base in an assignable ratio, but also when the fluxion of
the base being given, the fluxion of the ordinate of any subse*
quent order is to the fluxion of the base in an assignable ratio*
For in all those cases the point E is a cuspis of the sedond
kind ; or, if it is a cuspis of the first kind, the tangent at E if
not parallel to the base.

277* What we have said of the greatest and least ordinates
is easily applied tp-the greatest or least rays that can be drawn
firbm a given point to a curve. Let S (Jig. 102ybe a point that
is given in the plane bf the curve DPE> SA a right line given
in positiofa^ SP any right line from S that meets the curve in P,

PM

934 Of the points of ccfntrary Flexion Book I.

PM^ a perpeodicular from P on SA in M. Then^ the fiiixi<»i of
SM being given^ the ray SP is a maxinrnm or ndnimwn when
its first flujKion vanishes^ and its second fluxion does not ti^
nish, or when the fluxions of the subsequent successive orden
vanish, if the number of all those fluxions that yanish be an
odd number ; for if MQ be taken upon MP always equal t9
SP, the ordinate MQ of the curve IQ shall in those caries be a
VMxin^vin or mnimamt by prop. dl. When the nun^ber of the
fucc^ve fluxions of SP or MQ that vanish is an even num-
ber, the point Q is a point of contrary flexure in the curve IQ^
^d MQ or SP is neither a maximum nor minimum : but it does
not follow, that Pisa point of contrary flexure in Uie curve DPE.
fi7B. To illustrate by an obvious example the necessity of hav*
iog sqgaf d to thQ«e lin^jyuitic^s in resolving by the commo^i
rnles the problems that relate to the maxima and minima, let
DPE be a common parabda^ DK its axis, S a point given with*
m tl|e Gurye, SA be perpendicular to the axis DK in K, ai^d
PN p^ipendicular to it in N. Then, SM or PM being suppoa*
fd to flcfw vniformiy, let th^ fluxions of SP be computed. The
first fluxion of SP vanishes when SP becomes perpendicu-
lar to the curve^ and by the fommon rule SP ought in that
case to ^ a i^aximum or minimum. But it is certain, that if
DK be eqiifd to the sum of sDN and one half of the paranveter
of t^ axis, and the point S be not upon the axis, the right
line SP, though p^rpendiculiyr to the cun^e^ is neither a miuri-
imm nor nitiittw» ,* for a c^de described from the centre S
tbroui^ P M^ wijthottt the arch PD and within the arch P£,
as shall be demonstrated a^jwards. In this case it will be
found, ^bat the second fluxion of SP vanislies as well as its
ffsi (luxion, but that its third fluxion doies not vanish ; so that,
KPOidiHg to prop, dl, SP is not a maximum or minimum io
ibis case, though its first fluxion vanish. If the third fluxion
of SP also vanish, then SP is a maximum or minimum by the
wnne prcipo^ition. And this is thecase when the point Sis upon
the axis pit/and /D is equal to one half of the parameter; for,
inlliisc^Bse, when P comes .to D,the first, ^ond,and third ftax<-
ions of /P vani9h, but its fourth fluxion does not vanish^ as
will appear by the oomputation^ and it iskuown that/(>is the

least

Ohitp. IX. ^Hi4 R^siM. ^5

leaat right Ii|ie that can be drawQ from the point/ to fhe-para-^
bola. Nor >s there any curve but the circle alone that doea not
aflfiard eixamples of this kind. It may be of use for abridgiog
eompntatioDS of this sort to observe, that when the vahie of anjp
qnantity t is expressed by a term that is formed from ^^y powtr
of another qnantity u and invarible quantities, if whilf p pa fi«
Bite, the first fluxion of u and its fluxions of any subsequenMup^
ceasive ordersvanish; the fluxions ofz of the si^me orders viu^isd
at the same time. If jr be expressed by a fraction whose i»mi9#
rator N and denominator D are both finite wh^ Uie ^uxifil 9f
N is to the fluxion of B as N is to D^ th^n the /Siuxk^ of ji
vanishes ; and if the fluxions pf N and J) of axiy subBequf^M suo^
cessive orders be to each od^er r^peotiv^ly in the ^twc ratiq fif
N to D, dien the fluxioiis of ;t of the sapi^e oarder^ vanish.

€79* Letusnowresumetfaeceiiistnie(MMic^tbeei}th{^«l^
and 2 l^h articles, where P the intersection of LP the taqgeitt (if
any curve LB and of SP the perpendicular from the giviM
point S is supposed to be always found in the curve PP£; 90%
where it is shown, that if the angle SPT be made equal to SLP^
PT shall be the tangent of the curve DPE. Let ST be a)waj9
perpendicular from S upon this tangent PT, and T be f^w^ya
found in the curve GTH ; and, if AS be any right imegivon
in positbn that produced beyond S m^ota PTproduced^b^onit
T in Q, die angles ASL, ASP, AST «h^ be in aritfaM^dpal
progression, and the angle AST shall be equid to tA$P— >ASL»
The angle ASP being supposed to increase, the angle SQT i»*
creases or decreases according aathearcb of tbecunne deseiihed
by P is concave or convex towards S; and, because AST (os
SASP*-ASL) is equal to SQT added to the right «^g]e STQ ;
it follows^ that, the fluxion of the angle ASP being aupposod
positive, the arch described by P is concave or ooi^^mx towMvb
S according as the fluxion of £ASP — ^ASL is positive or iiega«
tive,thatis, according as thefluxion <^2A9P is more ctrleasthaa
the fluxion of ASL. When the arch described by L is 9oaveK
towards S, ASL decreases while ASP increases, the fluxionof
ASL is negative, the aaof^e ASTinoreaaes, and thearch dsaanb^
edbyP is always concave towards S. These things wens mmi**
tioned ainive^ but without apcoof*

S80. Whea

L_.___

1

tSS Of the Paints ofamtrafy Flexion Book!.

280* WhenthepointP(/^.104)inthecurveDEisapoint of
contrary flexure, the point T in the curve GTt is acuspis^ the
^ngle AST (or £ASP — ASL> is a maximmn or minimvan; and
Ifaerightline ST isalsoamaxtntttm oxminimumj unless when the
angfe^SPT or SLP is a right one, or when S and P coincide.
For lei the angle STK be made equal to SPT, so that TK may
lie the same way from ST as PT from SP, and TK shall be
|he tangent of the curve GTt at T, by art 211 and 212.
LeiPp be the arch terminated at P that is concave towards S,
and Pp the arch that is convex towards it ; and while P de-
acribes the arches Pp, Pp, let T describe the arches T/, Tt. It
is manifest, that these arches Tf, Tt are on the same side of
the right line PT; and, since they have the same tangent TK»
the point T is a cnspis. The «ngle SQT is a maximum or mi'
jiimum when P is a point of contary flexure, and therefore
ASl for 2ASP — ^ASL) is a maximum or minimum. When the
;«ngle STK or SPT is not a right one, the right line ST either
exceeds the right lines that can be drawn from S to the adjoin-
ing parts of .either arch Tt, Tt, or is less than them, and there*
Sore ST is a maximum or minimum.

.281. It follows, conversely, that the curve pp being conti*
nued on both sides of the right line PL, and the tangent at P
being oblitpie to SP, if ST heamaximtim or miniinuniy the point
P in the curve DPE is a point of contrary flexure. Therefore
/P an arch.of a circle described from the center S through P
.being suppofie4 to flow uniformly, if the fluxion of ST vanish,
the pointP is a point of contrary flexure, the same limitations
being understood as were described in prop. 2 1 . Let the inva-
riable fluxion of the arch/P be represented by a given line PI
taken upon PL, and let IH perpendicular to PI meet TP pro-
duced in^ H ; then, shall PH and IH represent the fluxions of
the curve DP and ray SP, by prop. 16 and 17. The rectangle
ocmtatned by SP and PI is equal to the rectangle contained by
STaadc PH ; and, therefore, when the fluxion of ST vanishes,
the rectangle contained by IH (which measures the fluxion of
6P) and PI is equal to the rectangle contained by ST and the
right tine which measures the fluxion of PH, and the fluxion of
PH is to the fluxion of SP as PI is. to ST, or as PH is to SP.

But,

'Chap. IX. a9id Rejkxion. S37

Bat, becauae PI is supposed to he invariable, it follows, from
prop. 15, that the fluxion of IH is to the fluxion of PH as PH
is to IH. Therefore the fluxion of IH is to the fluxion of SP
(which is expressed by IH) as the square of PH is to the reci*
angle contained by SP and IH; and, consequently, the right
line which measures the second fluxion of SP, PH which mea-
sures the first fluxion of the curve, and the ray SP are in con-
tinued proportion; which coincides with one part of the rule
tliat is usually given for finding the points of contrary flexure
in curves, that are considered as spirals, and are defined by an
equation that expresses the relation of the fluxions offP, SP,
or DP to each other. But this rule is not to be admitted wi^th-
out the limitations that follow from prop. 21, though the first
fluxion of ST vanish. If the curve described by P be not con-
tinued on both sides of the right line PL, or if the fluxions of
ST of the subsequentsuccessive orders vanish, and thenumber of
all its fluxions that vanish be an even number, we cannot con-
clude that P is a point of contrary flexure. When the first flux*
ion of ST vanishes, its second fluxion vanishes also (the flux-
ion of the archyP being invariable), when the ratio of the se«
cond fluxion of PH to the second fluxion of SP is the same as
that of PH to SP ; and when the fluxions of PH of any sue-
oessive orders firom the first, are to the fluxions of SP of the
same orders, respectively, in the same ratio of PH to SP, then
the fluxions of ST of these orders also vanish.

£82. It follows also from the 280th arUcle, that, the curve
PP being continued on both sides of the right line PL, and the
angle SPT being acute, P is a point of contrary flexure, when
the fluxion of the angle AST vanishes, or when the fluxion of
the angle ASP becomes equal to one half of the fluxion of ASL
(those cases however being excepted in which' AST is not a
maximum or minimum, according to prop. 2 1), that is, when the
angular velocity of SP about S is one half of the angular ve-
locity of SL about S, and is in the same direction; for if the
motion of SL and SP be in diflerent directions, the angular
motion x>f ST does not vanish when the angular motion of SP
is equal to one half of the angular motion of SL, but on the
contrary is equal to the sum of the motion of SL added to

ttviqe

SB8 Of the Points of contrary Flexion Book I.

twice the motion of SP. The point T in the cnrve GTH fa
a point of contrary flexure, for the same reason, when the flax-
ion of AST becomes equal to one half of the fluxion of ASP,
or when the fluxion of ASP becomes equal to two thirds of the
fluxion of ASL, because ASL, ASP, AST, arc in arithmeticai
progression. In general, the series of curves BL, DP, GT, ^c*
being continued, each of which is supposed to pass through the
intersections of the tangents of the preceding curve with the
perpendiculars from S on these tangents, that point in the last
cnrve of the series which corresponds to L in the first curve is
a point of contrary flexure when the fluxion of the angle ASL
is to the fluxion of ASP as the number of curves in the series is
to the same number diminished by unit; and that point is a cus-
pis when the fluxion of ASL is to the fluxion of ASP as the
number of curves is to the same number diminished by two,

£83. In this last manner the points of contrary flexureandre-
flexion are sometimes easily discovered (/fg. 105). For example,
when ALB is a circle, C the cehtre,Sany point within the circle,
AB the diameter that passes through S, the point P in the curve
DPiE is a point of contrary flexure when the square of DS is one
third part of the rectangle ASB, or (LS being produced till it
meet the circle again in Z) when LS is one fonil^ part of LZ.
For, let CV be perpendicular on LZ in V; and,SLRbeingmade
equal to CLQ, if LR meet the diameter in R, the fluxion of
ASL shall be to the fluxion of ACL (or ASP) as CR is to SR
(by .prop. 18), or as LV is to LS; and, therefore, when LS is
equal to one half of LV, or one fourth part of LZ, the fluxion
of ASP is one half of the fluxion of ASL, and P is a point of
contrary flexure; that case being excepted wherein CS is equal
to SB, and SB is one fourth part of AB, in which the point L
coincides with B when the fluxion of ASP becomes equal to
one half of the fluxion of ASL, and the curve described by
P is not continued on both sides of the tangent at B. When S
is nearer to the centre of the circle than to the circmnference,
or when S is without the circle, or upon the circumference, the
fluxion of ASP never becomes equal to one half of the fluxion
of ASL. When S is without the circle, the angular velocity
of SP may become equal to one half of the angular velocity

of

6i SL; biit the ttiotions of SP and SL are in opposilfe clirec*
tions when tkin happens^ the fluxion of ASP being suj^ioBed
positive^ the iSa^itiOn of ASL is negative^ and the finxion of AST
does not vanish; The curve ALB being stiQ a elrde^ and ibe ste*
ries of curves ALB^ DPE, GTH^ Sfc. bemg continued, th^ poiht
in ihe last curve of the series that corresponds to L in the circle
is a point of cohtmry fleiaire when SL is to S2 as the numbed
of curves diminished by unit is to th^ same number increased
by unit^ or when the square of SL is to the given rectangle
ASB in that ratio ; and that point is a cuspis when SZ is to SL
as the number of curves is to this number diminished by two ;
the case being always excepted when S is so situated that the
point L coincides with B when this happens.

284. In like manner these points are readily determined in
many other carves^ especially when the curve BLls such that
a point can be assigned^ from which rays being drawn to the
curve and perpendiculars to the tangents of the curve> the an-
gular velocity of the ray about that point is to the angular ve-
locity of the perpendicular in any invariable ratio : when BL
is any curve of this kind, it has no point of contrary ^xnre;
and if S coincide with that given point, the curves BPE, GTA
being of the same kind, * have ako no poi&t of conti^ flex-
ure ; but if S be any other point in the plane of tibe curve
BL, the curves DP£, 6TH, S^c. may have points of contrai^
-flexure and feflexion that are often easily determined, by ail.
fi8£ and 210. When BL (fig. 106) is a parabola, and S is within
the parabola upon the axis,Pis a pointof contraiy flexurewhen
LQ is equal to LS, or when BM (LM being perpendicular to the
axis in M) is one third part of BS. The curve described by
T in this case has two points of reflexion corresponding to the
two points of contrary flexure in the curve DP, and a third point
of reflexion upon the axis of the parabola at a distance from S
towards A equal to one fourth part of the parameter.

285. The right line ST (fig. 104) is also a maximum or mtW-
mum, if the fluxion of/P vanish when the fluxion of ST is sup-
posed to be assignable, by art. 245 ; and in this case the ratio 6f
the difl^rence betwixt the rectangle PHI and the rectangle

* DtfcHpt: CQfnrum^ p«rt 2. prop. 14. 16. Src.

con-

1

MO Of the Asymptotes of curve lines. BookX

ooQUined by SP and the right lUxe which measuxtes the fluxion
ofl^H to thesquaieof PHbjincreaaiqgbeciHnes unassignable;
and this coineides with the second part of the rule that is com-
jnonly given for finding the points of contrary flexure in eurves
that are considered as spir^. It is however to be allowed with
limitations analogous to those above-mentioned in the £45th,
'£63d^ and £67th articles. But having insisted on this subject
at a sufficient length, we proceed to consider some other af*
fections of curve lines*

CHAP. X.

Of the Asympioies of curve Lines, the Areas bounded by
them and the Curves, tlie Solids generated by those Areas,
of spiral Lines, and of the Limits of the Sums of Pro-
gretsions.

£86. A Ri^t Une given in position is aif Asymptote of the
branch of a curve when they never meet^ but iq>proach to
each other continually, so that by producing them, their dis^
tance from each other becomes equal to any line^ how small
.soever, that may be given ; and the branch of the curve that
•{^roadiies thus to the asymptote is said to be of the hyper-
boiic kind. A branch of a curve that approaches in the same
fanner to a parabola is said to be of the parabolic kind, and to
h^fve a parabolic asymptote, of which there may be as many
difierent kinds as there are parabolas of different orders. In
general, any curve lines may be said to be asymptotes of /sach
other mutually when they approach to. each other in this
manner.

287* In the common doctrine ofthe hyperbola it is shown^ that
a curve and aright line may continually approach toeachoiher
in this manner, and never meet. Let CO and CV (fig. I07)be the
right lines that are called theasymptotes of thehyperbolaaMEj,
aK, and IM two right lines parallel to CV bounded by the
asymptote CO and the curve in a and K^ I and M; and IM

shall

naleXI/i/.^4^J5V.A

9 9

Tifi.ff6

\zf :

^^7

P O

Fyiaoo.

Wlg.iai,

I^ao4'

U^.iod.

J^^JOZ.

I

Chs9« X. Ctfthe A^jpaptotez 9/ curve JUnes. 241

shaU be 10 aK as CK is to CI. Thereft«:e^ CK and aK being
given^ and CI being produced till it become any multiple how
great soever of CK, the ordinate IM is always assignable, be^*
ing the same part of aK as CK x»of CI f and IM continually
decceases, so that it may become less than any given line Z by
producing CI till its ratio to CK be greater than that of a K to
Z. When the ordinate IM is to aK as the square, cuKe, or any
power of CK that has a positive number for its exponent, is to
the same power of the absciss CI, the curve aME is of the
hyperbolic kind, and the same right lines CO, CV are its
asymptotes.

288. Let AD (fig. 108) be a right line given in position,^ a
given point,SPM any right linefrcHn S meeting AD in P ; upon
whichletPMbetaken always fromPequal to agiven lineAa:
then the curve aME shall be the conchoid of the antients, and
the right line AD shall be its asymptote. For the curve aM
shall never meet this right line AD, because PM is supposed to
be taken always equal to the given line Aa \ and as it nev^
decreases, 90 it cannot be supposed to vanish. But the curve
continually approaches Vo AD, and MN the perpendicular
upon AD may become equal to any right line Z how small
soever that may be given, by producing the figure; for if AQ
be taken upon Aa equal to Z,^» parallel to AD meet the circle
amx described from the centre A'mm, and SPM be ^rawn pa-
rallel to Am meeting Qm produced in M, the point M shall b^
in the conchoid, a^d MN equal to AQ shall be equal to Z.

289. To mention one other simple instance : whoever ad-
mits that a right line may be continued at pleasure, and that
any given right Une Jipay be divided into two equal right Ihies
(according to the principles of the common geometry), will allow,
that the base of a figMce beii^ produced, the ordinate may be
conceived to decrease in such a manner, that when the base is
increased by any given line, the ordinate may become one half
of what iyi/as before the base acquired that increment; in whidh
case the ordinate n0iw vanishes, because the half of a right Iw
is always a right line. JUt OA, AD, DE, EF, ^c. (fig. 44) h^
any equal right lines by which the base is successively in«
creased^; let Aa the ordinate at A be one ^alf of Oo.the ordi-

VOL. I. Q nate

»

2[4S Of the Asymptotes of curve Lines. • Boot I,

nate at O, Dd one half of Aa, Ee one half of Dd, and so on.
And it is manifest, that the logarithmic curve (described in art,
176.) which passes through the points 0, a,dy e,f and the ex-
tremities of ail the ordinates in the same series, which may be
continued at pleasure, can never meet the base. In general,
any quantity may be conceived to decrease continually, and yet
(fig I07)neverbe quite exhausted; as,whentherightliDeAPthat
touches the circle Agx in A is produced, and KP is always join-
ed, the arch org, the angle xKg, and the perpendicular g/ de-
crease continually, so that they may become less than any given
quantity of the same kind by producing AP to an assignable
distance, but never vanish.

• 290. We havementioned those simple instances, to show that
there is nothing so abstiiise or inconceivable in what Geometri-
cians demonstrate concerning the asymptotes of curve lines as
is sometimes represented. They are under no necessity of sup-
posing, that a finite quantity or extension consists of parts iitfinite
in number* orthat there are any more parts in a given magni-
tude than they can conceive and express: it is sufiident that it
may be conceived to be divided into a number of parts equal
to any given or proposed number, and this is all that is sup-
posed instrictgeometryconcemingthedivisibility of magnitude. .
It is true, that the nmnber of parts into which a given magni-
tude may be conceived to be divided is not to be fixed or limit-
ed because no given number is so great but a greater than itmay
be conceived and assigned : but there is not, therefore, any
necessity for supposing that number infinite ; and, if some
•may have drawn very abstruse consequences from such sup-
positions, geometry is not to be loaded with them.

• 291. Though Geometricians are under no necessity of sup-
posing a given magnitude to be divided into an infinite number
of parts, or to be made up of infinitesimals, they cannot, so well
avoid the supposing it to be divided into a greatertfmmber of parts
than may be distinguished isx it by sensein any particular deler^-
minate circumstances. But they find no difficulty in conceiving
this* and such a supposition does not appear toberepugnant to

• .. •

* Ffh\cip. of humui knowkdgr, § 1S4.

the

Chap. X# Of the A^fmptotes of curve, Unn. i4$

the common sense of mankind^ but on the contrary to be m»ost
agreeable to it^ and to be illustrated .by common observa^tioo*
It would seem very ifnaccountabje^ not to allow them to con-
ceive a given line^ of an inch in lengthy for example^ Viewed at
the distance of ten feet^ to be divided into more parts than are
discerned in it at that distance ; since by bringing it nearer^ a
greater number of parts is actually perceived in it. Nor is it
easy to limit the number of parts that may be perceived in it
when it is brought near to the eye^ and is seen through a Uttle
hole in a thin plate^ or when by any other contrivance it is
rendered distinct at small distances from the eye. If we con-
ceive a given line that is the object of sight to be divided into
more parts than we perceive in it^ it would seem that no good
reason can be assigned why we may not conceive tangible mag-
nitude to be divided into more parts tha ire perceived in it
by the touchy or a Une of any kind to be divided into any
given number of parts^ whether so many parts be actually dis*
tinguished by sense^ or not. If the hyperbola and its asymptote
were accurately described, they would seem to sense to join
each other, at various distances from the centre, according
to the different circumstances in which they might be per-
ceived ; but we may conceive the ordinate at the point where
they seem to join to have a real magnitude, in the same man-*
ner as we conceive a given liqe to subsist when it is carried to
so great a distance that it vanishes to sight, or any small particle
(a» an atom in the sun-beams) to exist, though it escape the
touch, or have no tangible magnitude. It may perhaps illus-
trate this, if it be considered^ that the curve cannot be said to
meet its asymf ^te in this case^ in the same sense that a circle is
said to meet its diameter, which it appears to intersect in all
cases, whatever the distance or position of the figure, or the
acuteness of the sense may be; whereas tlie ordinate of the hy-
perbola that vanishes to sight at a greater distance becomes vi-
sible at a less distance, and may be distinguished into more and
more visible parts, in proportion as it approaches to the eye, or
the sense is more acute. And, surely, it must be allowed that
^ere is groun((for a difference between a line tliat escapes the
sight and vnmhe^^ beeattse of its dirtance from the eye, and a

. Q S nne

^44 Of t^ Ait/tnptoHhy i'Ai ih Ai-tta Book I.

Ihx^ ihAt fn <io eds^ 6ah ever b« p^fi^in^d^ oi" ^ati b^ Siipposed
t6 Fravc dtiy fexwtence. Pferhfips it wift fee said by flome, that,
i»trict]y speakifig. It li not th^ sam6 X\tit tnsit In thod^ df^et'enl
6irMdi^Uift6^ has a greiit^r atid le^ /mtab^r 6f visible parts. In
afiswi^r to this, it is stiffici^nt for dur purpo^ t<5 6bs€tv^, that as
there cah hardly t)6 any Phil<3s6ph6r But trtll allb<f that there is
tome s^nse rn trhich it is ih^ same ifich-Iitte that has fnor^ \u
sible parts at tight ititrhes distan<^e frotn the ^ than wh^u it is
held at die length of the arm ; so rt is not iiicumbent on us to
^xplaminMiat sense this is to be nttderstoddaeeotding to every
scheme : it is enough that this se&se mast be supposed to be
^tarn and obvioiid^ as it is universal, and that Creometriciafis ought
to be allowed to consiiTer lines and figures hi this sefisc aswell as
every body else. Philosophers and the vulgar equally conceive
thesun axid plsmets, and the 6tlier objects of their Observation and
enquiries, to be the ^me bodies, when s^ti at different dis-
tances or different times : and if they were Act allowed to con-
sider those bodies as madfe up of more parts tlian are perceived
by sense, and Geometrietans were uhder the Same limitations as
to magnitude in general, they would toot be a little perplexed ;
nor is it the more intricate and subtile part of those sciehces
only that would be thus pared off. The learned author above-
mefttibned tells us, '' Hiat the magnitude of the object which
^ exists without the mind, and is at a distance, continues al-
^ ways invariably the same.** * He seems to speak o( tan-
gible magnitude. It is not our business here to eActnire iiow,
according to his doctrine, tangible magftitude iah be conceived
to exist without the mind, any more than visible magditude.
This concession perhaps is made only for the sake of his Ar-
gument in this place ; but the evidence for the existence of
such an object may very Well be supposed to approach to that
which we have for the existence of any other objects that are
not immediately perceived by us. And since he kdmits it,
and argues from it, in this treatise, it would seem that some
invariable magnitude is to be allowed,wlrich f we apprehend by

• New Theory rf VUioo, 4 65.
i Ibid, J 5*.

the

Chap. K. ^Q¥^M py thtm and the Curw. f4|

the )^gbt, though nqt immediately ; an^ tbi4 trW J^pM^
may he ^o^ceived tQ bf ^vidsd iQV> aoy ^v^i^ /^9hi^ <9l

f»x^, ficpm ih^ 4i^«AQi)^atipP9 Rr9P9^d l^ Qf^wm^i^imi^ w

tbi$ sul^ect. Jn ^^n^yU^ which, ft cnight \o be rei^eu^re^i
thatf s«rfiMW is wl cv^ioj^fdfjr^ bj tbm f|s a hody ^f tljif Jie^
seiKBihle magnitii^c^ b^ as ^^ ^rqiinatiQii or bouti^aiy $f f
body; a line is not confidered as a surf^pe of thf lei^ 9^imiVk
brea4thj but as H^ ter«unatiqfi or limit ^f a surfificf : i^or is §
Yoint consideied cms the lea^t seasxble UjQe> or a xnpmei^t a? ^
lea^t perc€^ptible t^m^ ; hut a point a^ a tfr^Uft^o;i^ of ft ]xf^,
and a moment «^ ^ termination or limit Qf tiine' In this s^M^
they conceive clearly ^hat a surface, line, point,*' an'4 a WQ**
mentof time is; aqcl i^e postftlata q{ Euclid h^ing ^owi^
and applied in this sense, the prpofs by which it is shown that a
given magnit^e may be conceived to be divided into any
giyen number of partp, appear satisfactory : apd if we avpij
tlie Hjpposi^g the parts of a given magnitude to h^ iji^finiiely
small, or to be ^finite i4 number, this se^ms to b^ fdl fl^al
>^ the most acrupulpus can r^ise.

£98. Opt to prpceed : the arch Jig (fig- 197), the pector AKg,
tmi the ordinate PM incnea^ (xmtin^ally whiW APis pro^aced$
hutthe arc|i i\^ never a^^o^nts tQ 42*, th^aector 4Kgtp th^sec<*
tor AV^j or the ordinate VM to tl^ gi^ren riight line AK tb^
di^ltaD^ betwixt the base AP and asymptote KI. An area
APNF my increase at the f fu»e rate as th^ arfil* Ag> the s^tor
AK^, us the oicdiftate PM inpraaiB^ ; a^d, by ftpwi^g in tha

same manner, it may approach in inagiiitufile t^ a given apacie
cp^ti^fii^ny while tbe Ag^^ if» jprod^ooi, a^fl^ver n^punt to
it li^m rfwnm^, fyx /in ejfarople, the ppfis|nwtipn of art. Sjil^

{fig^),m^Op}mmhms^ in U,QH m N,ON in K and^
fi», latjtiie leptrngites OpM, AaUS>, PafJC, £eKP.4!r^ he «:p«^
fk^, md left j^, J^e m»t A^ if) )L a^d V. The i«»cibang)eiKAH>
J)i, EK> 4r^- are lespitctwaly jequal *o the reci»«gles oa, ML,
JUVj ^fi* awJ tl¥e fl|ia» pf ikfin^ nactangles being sace^siveljr
re^^l iN> 9^, ph, 9V, Afi^ it k^ t^EdB9re aWay» lew Aan the

Q 3 tinned

• , • • • »

£46 Of the Asymptotes, arid the Areas Book I.*

tinued till their number become equal to any given number^
how great soever it may be. The differences betwixt their suc-
cessive sums and the rectangle o A are successively equal to the
rectangles AM, AN^ AR, ^c, which continually decrease, and
by continuing the series may become less than any given space.
Therefore the rectangle oA is the limit to which the sum of the
l^ctangles AH, DI, EK, S^c. continually approaches while the
figure is produced, and to which it never amounts, unless the
figure be supposed to be infinitely produced. As for the area*
bounded by the curve adef, and the base AF, it is always less
than the sum of the rectangles AH, DI, EK, S^c. axid therefore
is always less than the rectangle oA. It approaches to a lesser
limit, viz. the rectangle contained by Aa and Oo, if we suppose
o to be the point where the tangent makes an angle with the
ordinate that is half a right one; and the Kmit of the area
bounded by the curve oadef, ordinate Oo, and base OF, is the
tequare of Oo. For let any ordinate Vp meet oB in Z, and the
fluxion of Pj? (or of pZ) shall be to the fluxion of the base OP as
Pp is to Oo, by art. 176. Therefore the fluxion of the areaOPpo
is equal to the fluxion of the rectangle contained by Oo and pZ,
by prop. 4. and the area OPpo is equal to this rectangle by art:
94. And as pZ is always less than Oo, but approaches to it so
that the difference Pp may become less than any given line by
producing the figure, so the area OPpo is always less than the
'square of Oo, l)ut approaches to it continually, so that their
difference (the rectangle contained by Pp and Oo) may be-i
*Coihe less than any given space.

293. The rightline KI (fig. 107), parallel to the base AP,being
the asymptote of any curve aME, let PN, the ordinate from the
curve FNe, be always to the given right line DG, as the fluxion
of the ordinate PM is to the fluxion of the base, and let PM meet
4ip parallel tothe base in p. Thenthebase APshallbetheasymp*
tote of the' curve FN^, and the area APNF shall be always
less than the rectangle contained by Ka and D6, though th^
base AP and curve FN be produced to any distance how great
soever, but shall continually approach to that rectangle, so that
their difference may become less than any given space by pro->
ducing the figure. For it Appears, as in art. £48^ that the area

APNF

Ch«p. X. hounded by them and the Curves. 247

APNF is always equal to the rectangle contained by pM and
the given right line DG^ and therefore is always less than the
.rectangle contained by Ka and DG by a space equal to the
rectangle <;ontained by IM and DG^ which may become less
than any given space> but never vanishes; because IM may
become less than any given' right line^ but never vanishes^
since KI .is supposed to be the asymptote of the curve aM£*

294. When CV perpendicular to CO is also an asymptote of
the curve £Ma produced beyond a, the right line Cv (which
is the continuation of CV) is an asymptote of the curve de^
jscribed,by N continued on the other side of AF ; and> if the
ordinates PM, FN, meet the base on the other side of A, the
area APNF shall be still equal to the rectangle contained by
pM. and DG> and will in this case exceed any given space by
4;ontinuing the curve FN along the asymptote Cr, because
pM will exceed any given right line by continuing the curve
tf M along the asymptote CV« In the former case^ when the
x;urve FN was produced along the base AD^ which is one of
.its asymptotes^ the area APNF was always less than a certain
space (the rectangle' eontained by Ka and DG)^ which we
therefore call its limit In the latter case^ when the curve FN
is produced along its other asymptote Cv, the area APNF
may exceed any given space^ and has no assignable limit. They
who scruple not to suppose the curve and asymptote to be
infinitely produced^ say^ that the area APNF then becomes
equal to its limit in the former case, and that it becomes in-
finitely great in the latter case. And this area has been said iu
. certain cases to be more than infinite by some authors^ from an
analogy they imagined to be betwixt what is negative^ nothing,
and finite, and what is finite, infinite, andxnore than infinite.

29a. Butwhehthecuxve£Ma(^g. 109) continued on the other
side of A touches the right line CV in L, the right line Cv 13
still an asymptote of the curve described by N continued on the
other side of F, because the ratio of the fluxion of PM to the
fluxion of the base, or of PN to DG, may exceed any given
ratio while P describes AB; but in this case the ar^a APNF
. (which is equal to the rectangle contained by pM and DG) is
-always less than a certain assignable, space, viz. the re^rtangle

Q 4 con-

248 Of the JsymptoUs, &nd the Artus Book I;

contained by bh and DG (pa being supposed to meet CV m
b), because pM is always less than b\j^ In this case, the cum
FN being continued along both theaiBymptatesBD^Bo aCplea^
sure, the area bounded by the two drdinates PN, PN that ait
on different sides of AF, is ahrays less than the tectaagle ooa»
tained by CL and D6, to which it approadnahowev^a: as its
limit when the figure is proditced continuatty both w*ays^
And when the curve is supposed bo be infinitely produced both
ways, and the asymptotes BD^ Bo are also supposed to b^
both infinitely produced, the whole area included betwixt the
curve and its two asymptotes is said to become ecjual to this
rectangle contained by CL and DG.

296. While the point N (fig. 107 and 109) describes the
branch of the curve FN that proceeds along the base AP which
is one of its asymptotes, if Pnbe to DGas ttie fluxion of PN is
to the fluxion of the base AP, the area AP^ shall be always
equal to the rectangle c*ontained by DG atid the excess of ^F
above PN ; and the rectangle contained by AF and DG is the
limit to which the area APTjf eontimrtilly approaches by pro-
ducing the cur\'ej^ and base AP, And if a series of curves
be deduced m this manner (the ordinate ^«ny curve being to
the given right line DG as the fluxion of the ordinate of the
preceding ciarve in the series is to the fluxion ofthebaBe),the
xectangle coTitained by the ordinate at A of any curve of the
series and by the right line DG shall be the limit to which the
area of the subsequent curve continually approaches while Jt
is produced along the base. While the point N describes the
branch that proceeds along the asymptote Bt7,thepoiRtn de-
scribes a curve that has the same asymptote ; but liie ai^a AVnf
in this case may exceed any given space b}^ producing lire
curve ffij and has no assignable limit : and the same is to be
said of^jhe area of any stibscquent curve in the series.
297 .When the cmreaME (fig-\ I0)hasnotan asymptoteparal*
lei to the base AP, but the angle JfcMTformedby MT thetangent
at M and Mfc paraflclto the base decreases so thtfthy prodwing
the curve it may become less thun any given rectilineri «ngle,but
never vanishes, the base AP is fi*fB an Bsymptote of the curve
described by N, because iSie ratio of PN to DG (which is the

same

Cbtip. X. Jfomnded fy them mi the Curves. 849

same «B that of iT to MA^ by f rop. 14), may become less dMoi
any gmii mtio^ and yet net^r vuiiflhes. [a thm «afie theiareft
APNF may exceed luay giveii space by producing the xmrre and
base^ became pM may -exceed any given right line. WhenCV
is AA asymptote of the curre £M« prodoced on tlie i»iher side of
«, Cv is an asymptote of the curve described by N produced
on the other side of F; aad^ PN being on tlie other side of AF^
the area APNF m^^ «xeeed in this case ako any given space
by prodacing the curve idoog the asymptote Co. Th^sts, when
4iM£ ifi the logarithmic curve described in art. 176^ bV its
asymptote^ da the prdinate whosek>garithm isnothing^ the curve
FN is the common hyperboln^ because the fluxion of PM is to
the fluxion of BP (or Ap) as ^ is to BP (by art. 176), and,
PN being to DG in the same ratio, the rectangle BPN is equal
to the invariable rectangle conti^ned by bet and DG. It fd-
iows ftom the genesis of the logarithmic, that pM may «xeeed
.any given right line by prodnciBg the curve on either side of
Aa, and tbex^^(»e the aren APNF may exc^d any given space
by producing the hyperbola on cither side of AF : the same
a.ppears from the common doctrine of the hyperbola. Hie
curve /it (ait. 293), in this case, ^nd all the subsequent curves
of theseries, are hypeib(4as of an higher order ; and when P is
upon the same side of A with D, the area terminated by tbe
curve and base cmd tibe ordinates at A and P has an assignalAe
limit which it can never exceed, the sante that was defined hi
the hA article ; but when P is on the other «ide of A, betwia^
A and B, that area may exceed any given «pace by producing
the curve.

298. Hiecmiverseof the 293d axtic\e(fig. 107)easily appears.
Let thebase ADbean asymptote of the curve FN, and <rp paral-
lel to the base through any given point in the perpendicular at
A meet PNinp; let tlie rectangle contained by pM and the
given right Kne DG be always equal to the area APNF ; let the
rectangle ad (Kd being equal to DG) be the limit to which
the area APNF continaally approaches, so that their difference
becomes less thaa any given space by producing the figure, if
any such limit is assignable; and let oK be taken on the same
side of ap withpM : then aright line through K parallel to the

base

1250 Of the Asymptotes, and the Arca% IBook I.

base shcdl be the asjnsiptote of the curve aME. For the rect^
angle contained by pM and DG shall increase in the same man-
ner as the area APNF (to whidh it is always equal) increases :
and as this area approaches continually to the rectangle con-
tained by Ka and DG in such a manner that their difference bcr
comes less than any given space by producing the figure APNF;
so the ordinate j^M approaches continually to aK or fily and
their difference IM becomes less than any given right line by
producing the 'figure aMIK. Therefore KI is the asymptote
of the curve aME. If no such limit of the area APNF can be
assigned^ but this area may exceed any given space^ then the
branch aME is not of the hyperbolic kind^ and has not a rec-
tilineal asymptote. For in this case j?M may exceed any given
right line by producing the figure aMp ; and (kt parallel to
j}M being supposed to meet the tangent Mt in t and MA equal
and parallel to DG in U), since PN^ or kt, may become less
than any given right line^ the angle ftM^ may become less than
any given rectilineal angle by producing the curve.

299. When the ordinate PN is reciprocally as any power of
BP whose inde:2( is greater than unit^ a limit of the area APNF
can be assigned, and the curve aME has an asymptote parallel
to the base. If the ordinate PN be reciprocajly as the square
of BP, and the curve FN with the base AP be supposed to be
infinitely produced, the ordinate PN at an infinite distance is
^id to be an infinitesimal of the second order, because PN is to
AF as the square of BA is to the square of BP : and the ele-
ment of the base AP being supposed an infinitesimal of the first
order, then since the element of PM is to the element of AP as
PN is to DG, it follows (according to the doctrine of infinitesi-
mals) that the elementof PMin thiscase becomes an infinitesimal
of the third order. If PN be reciprocally as any higher power
of BP, the element of PM becomes an infinitesimal of an order
still lower^ AP being still supposed infinite: And it is proposed
as a rule,* that, tlie base being infinitely produced, if the ele-
ment' of the ordinate bccoii(ies an infinitesimal two or more de-
grees beneath the element of the base, then we may conclude

that

Chap. ^. ^ bounded by them andtlte Curves, £51

that the curve has an asymptote parallel to the base. But^ since
it is acknowledged that we may be led into mistakes by this
rule, unless regard be had to the ratio of PM to AP yrheh AP
is supposed infinite (as when aME (^g. 1 10) is acubic parabola
perpendiculartoaj7ina^ and the ordinate is suposed infinite^ its
element is an infinitesimal two degrees beneath that of the base,
because in this curve the element of pM is to the element of
op as a given square is to the triple square of pM., and yet
this curve has no asymptote) ; and when this ratio is known,
it may be discovered easily firom thence if tlie cur\'e has an
asjrmptote parallel to the base, we shall not insist on this rule
iurther, in this plrtce, but see art. 331. •

300. Let KO (Jig. Ill) parallel to the base be now an asymp-
tote of the curve FN^, aod the rectangle contained by PM and
DG be always equal to the area APNF, as in prop. 20. Let the
itctangleKR be equal to the limit to which the area FN IK Con-
tinually approaches while the curve FN and asymptote KI are
produced^ if any such limit can be assigned ; and let AR be
taken from A towards D when the curve VNe is betwixt the base
AD and asymptote KI, bqt in a contrary direction when the
asymptote KI is betwiit the base and curve FNe. From R to-
wards D take R^ equal to DG ; and, dh being parallel to
PN oh the same side'-of the base with PN, and equal to AK,
join RA and it sh^U be the asymptote of the curve AM£. For,
if PN meet Rh in S, PS shall be to dh (or PI) as RP is to
.Rrf or DG, and the rectangle contained by PS and DG equal
:to 'the rectangle, RI. Therefore the rectangle contained by MS
land DG is equal tp the difference betwixt the area APNF and
the rectangle Rl,.or to the difference betwixt the rectangle RK
and the area FNIK. But this difference decreases continually
.while the figure is produced, and may become less than any
given spaee by the supposition. Therefore MS may become less
than any given right line; and RS is an asymptote of the curve
AME. But if the area FNIK may exceed any given space by
producing the curve and asyinptote, then AME is not a branch
of the hyperbolic kiAd, and no right Une can be assigned for
its asymptote. The angle however which the tangenJt at M
fonofi with the ordinate PM approaches continually to the angle

KA*,

n

25S Of the 4P/n9tofe$, wd the 4reaz Book L

KMi Kit>^u»g t^km iipoft KI eq^ol to DG* And we ve not
to cojaclude tbut tb^ curve has always |tn ajiymptote^ wben^
accoiding to the lungwge ef ihom who employ in&iitc;^ and
infiaitettmids iq tbb dootfine^ 4)e .a;urve ha$ an infixutp branch
ihat 9i ite tenoiMtU^ bec^m^s oblique to the base.

301. It 19 manifest;, i^onversely, that if RS be an aqrmptot^
of the evnre AM£(» FN he to the given light line BG as th^
flnacion of PM ie to the fliaion of AP ; and, Rd equal to DQ
being taken from Ein the same direction as P is from A>if(tt
paialiel to PM mftet RS in k^ then a right line through k par
rdlel to the baae $bajl be the asymptote of the curve FNf ;
and the rectangle RK ;5hall be the Umit of the area FKINj,
fcetsg always gitiato^ than this area hy an excess that decreases
nwitinnflHy^ and becomea kes than a^y given space by prpdyo^
iiig the figure*

d02. The eontinnisiio?!! of <bo0e theorems will appear Uovf
the £dUowing pEoposition^ aad its eonvwe.

PROP. XXI\^

Let the Hut Bm (Jfg. ll^S) be an aufnuptote of any kindeftkt
curve PM ; the Jluxion of ike base being represe9Ued by ih
given right line DO, let the fiu scions of the orOnates PM,
Pw he always ^measured by the right lines PN, P« ; mid,
when PM is equal to AF, let Pm, PN, and Pn be e^al €»
AB, Af, and Ah, respectively : then the rectangle contaiaed
by BF and DO shall be the' limit of the area bfUn.

ThccTirre Bm is an asymptote of Ae curve PM (art. £86),
'ti^en yim contimmllydecrcascs, so that it becomes eqiaiJ to any
Ti^ht line how small soever that may he given by .producing the
figure, hut never vanishes. Let FK, BL,/A, and bl parallel to
-the base meet PM in K, L, *, and /, respectivdy. Bccanae PN
IS to DQ as the ^uxion of the ordinate PM is to the fluxion of
•Che base AP, it fdlows, that the fluxion of the area APN^is
e^i^al to tibe£uxion t>f iiiei^cti^gle -contained by DG and PM .

There-

Cliap. A. bountted by thm and the Curves. i5d

Therefore the at^ff AFKfis eqnttl t6 th« /eouftgle conUiMd
by D6 and KM, by tbeor. 4, tAnt^ AP and KM begin to b^
gen^ated at the satll^ tim^. In the dame ttannerylh^ ar^ APnb
is equal to the rectafigle contained bj DO and hm^ AM, t<m^
tfequently^ the area ^f^n (the difference of AP#** and AP^f)
U i^tial to the rectangle contained by DG and the diifet«ii0S
bet^xt Lm iind KM, or the excess of BF aboVe Mm. Th«re^
fore th& area bfStn is always les6 than th^ reetangle eofltained
by DO and BF ; but it continual^ approached to thid reotangk
as its Utility tsilice Mm continually decreases, and becomes lesd
that! any gitien right line by producing the flgare. We hav«
supposed the lines Bm and FM to be on the sam^ side ^ AP ;
but the delnotistratioti is easily adapted to the othey eased, tt li
obvious ho\^ever that this proposition catinot be extended to
ih€ dasfe when the point m is fbtmd in a rectifineal asymptote of
the cthre FM that is perpendicular to the base ; tor AF and
l^M never meet this asymptote, being pardtel to it.

303. Cor. I. Since Mm the difference Of PM and Pm c&a%U

«

hually decreases^ and becomes less than any given right line by
producing the figure, but never vanishes, it follows, that Hm
which nieasures the deference of their fluxions continfually de-
creases, and may become less than the given right line DO
which measures the invariable fluxion of the bade in any given
ratio : and therefore bn is an asytrrptote of the curve^^N.

304. Cot.ll. llteCDnverse of this proposition easily appears:
that if bn be an asymptote of tlie eurve/N of /my kind, and,
the point F being taken any where upon Af the perpen-
dicular to the base at A, BF be tak^n from 1^ the contrary
way that bfh (vomf, so that the rectangle contained by D6
iand BF be the Ihnit of the area ft/N^ ; and if the rectangle
contained by KM and D6 be always equal to the aipea APN/i
kud the i^ctangle cotitamed by X^ and DG equal to the area
A9nb : th€n the curves 'Bm and FM shall be aliymplotes to
each other mutnally. For the rectangle contained by Mm and
DCr shall be equal te the e^cce^ of the rectangle contained by
BF and DcJ above the area hfUn ; and, since this excess may
become hsA than any given «pa<5e by continuing the figure, it
follows, that Mm jnay become less thah any given right line.

But

£54 Of the SoKd generated by the Area Book !♦

But if the area Jjf*N« may exceed any given space by producing^
the figitire^ then Bm is not an asymptote of the corve FM«

305. Cor. III. When Bm is a right line parallel to the base,
Vn vanishes^ and bn coincides with the base AP, which there-
fore is the i^ymptote of/N, as in art. 293. When Bwi is a right
line oblique to the base^ Vn is invariable^ and bn is a right line pa-
rallel to the base, as in art. 301 (fig. 1 13). When Bm is a com-
tnon parabola that has its axis perpendicular to the base, bn is a
fight line oblique to the base, and js tlie asymptote of the curve

jfN : and if the right Une bn be given in position, and the li-
mit of the areai/'Nn be known, the parabolic asymptote Bm is
determined by taking BFso that the rectangle contained by it
and D6 may be equal to tliat limit producing nb till it meet
the base in R, and upon RI parallel an(l equal to AB taking
IE from I the same way as A is from by so that the rectangle
contained by IE and DG may be equal to the triangle AKb ;
for E shall be the vertex of the parabola required : and if RV
be taken upon RI equal to 2DG, and VA parallel to the base
meet R6 in A, then V/i shall be the parameter of the axis of
the parabola.

306. Cor. rV. When Bm is a cubic parabola, bn is a com-
mon parabola ; and when Bm is such a parabola that the ordi-
nate Pm is as a power of the absciss whose index is m, then P»
is as a power of the absciss whose index is less than m by unit ;
and when m is less than unit, bn is some hyperbola that has the
base for its asymptote. , But we have insisted on this subject
sufiiciently.

307« As a right line, or area, may increase continually, and
never amount to a given right line, or rectangle ; so a solid may
increase continually, and never amount to agrven cube or cylin-
der. LetKI(/fg. 1 14) parallelto tbebaseAPbethetisymptote of
any curve aME, as in art^. 293, and, K(2 being taken upon KI
equal to any given right line D6, let dZ parallel to KA meet
the base inZ, and Mx parallel to the base in g, and complete
the rectangle augx. Let the square of PN (the ordinate from
the curve FNe) be always to the square of DG as the fluxion *
of PM (the ordinate of the curve aME) is to the fluxioi^ of
the base AP. Then the solid generated by the h}rperboUc ai^a

APNF

Chap. X. bovnded hf the A^mptote and Curve. $55

APNF revolving about the asymptote AP shall be equal to the
cylinder generated by the rectangle ag revolving about the axis
aK ; and the cylinder generated by the rectangle ad about the
same axis aK shall be the limit to which the solid generated by
the area APNF shall continually approach while the figure is
produced. For^ since the square of PN is to the square of D6
{orgx) as the fluxion of PM(or of ax) is to the fluxion of
AP^ a cylinder upon a circle of the radius PN of a height that
measures the fluxion of the base AP^ is equal to a cylinder upon
a circle of the radius gx of a height that measures the fluxion
of ax* Therefore the fluxions of the solids generated by the
area APNF about AP, and by the rectangle ag about aK, are
equals by prop. 6^ and these solids themselves are equals by theor.
4. The right line ax is always less than aK, but approaches
to it continually, so that their difference IM becomes less than
any given right line by producing the figure; and therefore the
solid generated by the area APNF about the asymptote AP is
always less than the cylinder generated by the rectangle ad
about aK, but approaches to it continually as its limits so that
their difference (the cylinder generated by the rectan^ Kg
about Kx) may become less than any given solid by producing
the figure.

308. Let the square of Pn (the ordinate of the curve fn)
be to the square of DG as the fluxion of PN is to the fluxioa
of AP, and the cylinder generated by the rectangle FZ about
FA shall always exceed the solid generated by the area APnf
about AP^ and be cqnal to the limit to which this soUd ap-
proaches continually while the figure is produced ; for if nq
parallel to the base meet dZ in q, the soUd generated by APn/*
about the axis AP shall be always equal to the cylinder gene-
rated by the rectangle Fq about FA. If this series of curves
be continued in the same manner^ suppose tfae-rectangle that
has the ordinate at A of any curve in the series for its base and
its height equal to Az (or DG) to revolve about AK^ and the
cylinder generated by it shall be the limit of the solid gene-
rated by the area of the next curve in the series revolving
about the asymptote AP.

309. If the curve EMa(/?g. 1 15)producedontheothersideofa
touch BV in L^let LR parallel to APmeet KA and (2Z produced ia

k and

£59 Of th^ Solid gemtrated by the Ana. Book L

k aad R>aiid tbe Bolkl generated by APNFaboat AP when P is
betwixt Aand B, shall be always leai than thctcylinder generated
fay tberectan^ diRieTolviDgabonttheaxiiaft. Btttif BV(/{^.
1 14) ia an asymptote of the curre EMa^ tha solid generated by
APNF in this case may exceed any gtren soiidby ^oducing
tbefigme.

310. When «M£ is a common hyperbola, CO and CV its
asynxptotes, the curve FNe is also a common hypeibolaj and
BA and BC its asymptotes. For in this case the fluxion of PM
is to the flnxioo of AP (or the fluxion of IM to the fluxion of
CI) as IM is to CI, or as the rectangle CKa to thef square of
CI ; and, by the supposition, the square of PN is to the square
of DG as the fluxion of PM is to theflnxion of AP : therefore
VS is to DO as a mean proportional betwixt CK and Ka is to
BP. Ftom which it follows, that the solid generated by the
hjrpertMdic area APNF about AP is always equal to the cylin-
der generated by the rectangle ag about the axis aK, and never
amoonts to the cylinder generated by the rectangle ad about
oK, or BF about BA, but approaches to it as its limit. But
if AP be Uken from A towards B, the soUd generated by APNF
may exceed any given solid by producii^ the figure. It appears
in the same manner, that if PN be reciprocally as any power
of BP whose index is any number that exceeds |, then a cylin*
der may be assigned that always exceeds the sc^d generated by
APNF about the axis AP, the figure being product to any dis-
tance how great soever. WhenaMEO^g. li6)istheIogarith-
Biic curve, attd KOite asymptote, FNe is also a logarithmic, and
AP its asymptote. Fear, aince the fequare of PN is to the square
of DG as the fluxion of PM is to the fluxion of AP, or as IM
to aK (supposing nK to be the ordinate whose bgarkhm va-
Bishes, and is equal to the invariable subtangent of tlie curve) ;
it follows, that the fluxion of PN is to the fluxion of IM as PN
is to 2lM. Therefore the fluxion of PN is to the fluxion of AP
as PN is to ^K. Hence the cylinder that is the limit of the
solid gencsated by the area APNF in this case has its axis ulU
equal to one half of AT the subtangent, and the radius of the
base JUL equal to the ordinate AF. The continuation of the
daeofems of this kind will appear from the following propo-
sition.

PROP.

Chap^ X. hounded by the Asyn^Mt aid Curve. 9ST

PROP. XXV.

3\l. The rest remaining as in the fast proposition (Jig. 112)i
suppose ngw that the square of PN is to the square ofJ)Q as
the fluxion of PM is to the fluxion of AP, and the square of
Pn to the square of DG as the fluxion of P/» is to theflux^
ion of AP : then, if the area bfNn be supposed to revolve,
about the axis AP> the solid generated by it shall be always ^
less than a cylinder upon a circle described with the radius
BG of an altitude equal to BF, but it shall ^j^roach conti-^
finally to this cylinder, while the flgure is produced, as its-
limit.

It is demonstrated^ in the same manner a»in a|t. 307y that if
B(2 parallel. to AP be. equal to DG^ and Fx be taken equal to^
Mm from F towards B> Uie cylinder generated by the rectangle
dx revolving about Br shall be always equal to the solid gene-
rated by the area 6f Nik about the axis AP. Therefore the cy-
linder generated by the rectangle F<j about the axis FB is the
limit of the soUd that is generated by the area ^Nn about AP.

312. In art. 301 (/g. Ill) let MT touch the curve AM in M,
and meet the base in T; and^ while the curve AM is produced
the point T shall approach continually to R^ the angle PTAI
to the angle PRS, and consequently the tangent MT shall ap-
proach continually in position to the asymptote RS. For^ let
PYbe equal to the given right line DG, and Y/ parallel to PM
meet KI in I, and NL parallel to the base in L ; and MT the
tangent of the curve at M/shall be parallel to PL, by art* 247.
The right line P/ is parallel to the asymptote RS, by art. 301.^
Therefore the angle LP/ is equal to the angle formed by the
tangent MT and asymptote RS : and, since li (or IN) may be-^
come less than any given right line by producing the figure while
PY remains invariable, so that the angle LP/ may become less
than^any given rectihheal angle; it follow^, that MT approaches
conthiually tO the asymptote RS iti position. Hence, when

VOL. I. R the

258 OfdtUtmmng At Tangents Boiq|k I.

the curve is supposed to be infinitely produced^ the tangent at
its infinitely distant termination is said to coincide with the
asymptote ; which therefore is <?omi|ionIy considered as the
tangent of th« curve at an infinite distance. In the same
manner, when two curves are asymptotes to each other
mutually, they are said to touch each oth^r at an infinite
distance; ;and, conversely^ whei) hoth ^re supposed to
be infinitely produced, if they are such as must be supposed to
meet at their termination, they are s^d to be asymptotes of
each other.

did. Hence the theorems for drawing tangents to curves
often lead to such as serve for determining their asymptotes,
there being no more required than to find thq right line to
whieh the tangent approaches in position, while the curve is
produced continually. Of these we shall subjoin one, that fol-
lows from prop. 18, by which the asymptotes are determined
when a curve is supposed to be deseribed by the intersection
of right lines that revolve about ^ven points, and the ratio of
the velocities with whi(& those fines revolve can be found.

PROP. XXVL

The rest (Jig, 117 and 118) revu^iuing « in prop. 18, let the
fluxion of the angle ACP be to tic fluxion ^ aSP (or the
angular velocity of CP about C to the ULnguiar velocity of
SP about S) while CP and SP become paralkl, as SQ « Xa
CQ; and^iJfKK the asymptote oftht curO^ described by ?
meet CS in R, CR shall b^ equal te SQ.

It follows, from prop. 18, that if PD l>e the tangent of the
curve described by P, the w§te SPT b^ iftade equal to CPD
according to that proposition, «a4 FT meet (^ in T ; the
fluiuon of the angle ACP sbaQ be tQ tbe flunOA of the angle
iiSp (or the angular velocity of CP tQ tha m^gsi^T vdocity
of SP) as ST is to CT. UK tlie wgte PCV be alvajs equal
to PST, an4 PDm^t SC apd CV i© P aod V j let Gr
and Sy be parallel to the asymptote RX, and the i^pg^ ^if

being

SP{ and the ratio <tf C V V> ST qipv^wiMM emtiuudiy to. il ^
tio of equality yMk P desmW tbe branqk of tka qiu^ that
belongs to the oagmptote RXf beomui^ while th« rigbt lii^fi CP,
SP approach to parallelism^ tbtir ratio approaches to a ratio of
equably: btttCoistbelimUtoinhiobCV«ppr<M|ches«%thesaime
time, because (by the last nrtiole) the tiqagontifD approaeben m
positioa to tbe asymptote RXi CP to Cjf, ani) tbe an^ PCV
(or PST) to^, or XRC, or j:Qo. Tbe ^iigle (^vR b^iiis
equal to CRo, Cf^ is equal to CR ; ^nd therefore CRia^foiillo
the limit to which ST eontwnaUy f^pronches^iiePdtaoriiks
the conre FH. Bet mnm tbei nitip of the fluicion of tbo apgfe
ACP to theiesioiiof ^^ when CP and SP become panilH
is that ofSQ to CQ^by the supposition; it follows, that $Q is
tbe Umit to which ST approaches eo^tiimaOy while P desqribes
FH.TherefoieCRiaeqMltoSQ. Iftbe angles PCS 0(^.118),
PSC increase or decieoae togetberwhUe CP aod SP become pa*
rallel, tbe point Q mmit be tafcso npo«i.C9 betwixt C and S;
bat itis tobetakeniqponCSpiodiiqedbqroftdCC^f. 117)> orbe^
yond S^ when one of those amgles deoreasef wbij^ the oiher in^
creases, acooidtng aa liie Telooi^ of SP aboDt Sis greater or less
than that of CP abtat! C: and it is moaifest, tbot GR md SQ
must be ahraya taken in contrary dlreetaoosirom C and S. If
one of tbe angles PCS, PSC d^cp^ease^a^ tbe other increase
while the point P describes FH> and |hei««ions of those angles
be equal wheaCP asMl SP beeonsd parajfeli theponpit Q is not
thenassignaUe^andisssidtobfoonleinAintdy distant; in.which
casethebranchof tht cumedescribedby P isnotottbehj^ibolic
kind, unless somethoea whea CP coincides with CS»

314. C0f. Let SQ be to CQ a# the angular velocity of CP
is to the aagnlar velocity of SP M th^ term er moment when
diose lineabeeomo periUe)^ and let CR be tai4m eqtal to SQ
with the piecaistions we have dpscijhod ; then RX drawn pi^
caBelto CP or SP (which are snpposod pnraUejl to each other)
tirnH be an asymptote 4»f tbe curve described by P« If the
sight lues CP and SP be always tijugents of any oerve lines
that pass thsoogh G and $, iniifoiul of revolting about ^ko$t

R4 points.

<60 Of dethrmimng t\e Tangents - Book 1,

points^ the congtraction by wKich the tangent or asymptote of
the curve described by P is determined; is the samew • The use
of the 1 8th proposition for drawing the tangents of curve lines^
aiid of thispropositionfordeterminingtheir asymptotes, wiU ap-
pear from the following examples^* which we chuse from a great
number that taight be brought: *

315: Ex. 1. The righttinesOA^Sa beitig given in position^
let the- angle' AOP be always to aSP in atay given ratio ; deter-
mine the point T so that ST may be io'CT in the same given
ratio^ and thatT may lie in the same or in an opposite direc-
tion from C rtnd'S, according as the right lines' CP, SP revolve
about C and S with the same or with opposite directions; join
PT, and make the angle CPN equal to SPT, so that those
angles may be on opposte sides of C7P and SP: andPN shall
be the tangent of the curve described b^ P. The point T is
a given or invariable point, because the ratio of ACP to aSP is
supposed to be given ; let this ratio be that of m to n, and ST
shall be to the given fine CS as m b to the difference of m and
« when CP «nd 6P reVolve about C and S in the same direc-
tion, but as m is to thfe dm^iofni aUd n when' they revolve in op-
posite directions. ' Let CRbe taken from C equal to ST in an
opposite directtoh; and tlie asymptotes of die curve (tf ithas any^
that is^if 'CP andSPeverbeoom^parallel while liie curve is de-
scribed) shall pass thhnighR {fig. 119). To mention afewof the
simplest cases tWatcu'e comprehended in this example, let ACP
be equal to aSP, and if CPand SP revolve about C and S with
the same direction, the point P shall'^describe a circle : in this
case PT is to be considered as parallel to CS, and die angle
CPN being made equal to SPT or PSC, PN is the tai^ent of
the circle, as is well known. ButifCPandSP(/g. 1£0) revolve
about Cand S with opposite directions, the point Pshalldescribe
an €qnilateral hyperbola. For if Sa be a right line given in posi-
tion, ASP be always equal to SCP,PH parallel to SA meet CS
in H ; the triangles HPS, HPC being similar, the rectangle
SHC shall be equal to the square of PH, and P shall describe
an equilateral hyperbola, of which CS is a diameter, and SA a
tangent at S. Bisect CS in Ty joint PT, let the angle SPN be
made equal to CPTy and PN shafl be the tangent x>f the hy-
perbola

Ch^.X. end AsympiiUii. flSl

pierixd^ at P : as. in the cir<de, if srN be made equal to CP^^
t being the ceotie, and CS a diameter> PN is a taogeut ;.onIy
ttie aogle SPN in tbe circle is taken the eonitcary way. . Sow^
other analogies betwixt the circle and equilateral hyjperbola'apr
pe^r alaoftomthis: 1. IietCS(^jr.n9iii)tdii<20)aiiydiaiaeteT«oL
the hyperbola be drawn^ and right lines from CandStoPany
poiotintheh]q)^bQla9iftheangIePSAbeniad^eqHaltoPpS^$A
shallbethetangentatS; the tangent of tbecircleatS i^de^r^
min^ by thesame coQstruptioAjonly tbatsv)gleistpbe<tal^ir. ^a
ti^o<>the;r«idet)f SP. S^ Wbenajiytwp pcMpts*Pand/)are.ip the
equilateral hyperbola^and right lines are inflecj^ fiomC andSj
tbeextremities of a^y diameter CStliatpnsse^ through thec^Qtvet
T, to those points, the angle PCp is equal to PS/^i or to^its $up^
plement to two right pnes. 3. The angle PTS contained! byt
TP, TS any two semidiam<^£8 of ^the hyperbola ,is eqaal |o
the angle PVA contained by SA> PN tbetangents at theextre-
mities of those semidiameters : for produce PT till it uu^tthe
opposite hyperbola in K, join CK, CS ; and, TV whicli is pa*
rallei to CP (because it bisects CS and SP), and the angle PTV
being equal to CFT or SPV, and STV equal to TSK or PSy,.
it follows, that PTS is equal to PVA. As for the s^yipptotes^
because the angular velocity of CP is equal to the angular ve-^
locity of SP, it follows, from the last proposition, that they pass
through T, the point R coinciding witli T ; and, if PB, P& be
taken eq^al toPS,the asymptotes shall be parallel to^B andS6,
because, when Sp comes into the position of SB or Si, then Cp
and Sp become parallel.

3 16. If the angle PSC (^g. lGl)be alwftysdouble of PCS, take
STfrom S towards C equal to one third part of SC; andthepoint
P shall describe an hyperbola whose transverse axis shall be CT,
and that shall have one of its^bci in S : and, if the angle CPN.
bepiade equal to SPT, according to the last proposition, PN
shall be a tangent. Take CR equal to ST, or one third part
of CS,and the asymptotesshallpass through R constituting with.
RS an angle of 60 degrees, because CP and SP become paral«
lei when the angle PCS^^g. 122) is of that magnitude. Iftbean-^
gle PCB be always double of PSB (the point B being situated
on CS produced beyond C), it is obvious, that P shall de-

R 3 scribe

fiftK Of determining the Tmgentz Book I.

aetibe a oit^ pawing through S hating its omtre in CjMd that
Tiithe€jm«iiiilyofth^diamatArtharoii^h S (fig. 119)* IfCAand
S« fttaiaiiy given angles with CS^ACP beahrays doaUa<tf «SP,
and the ae^ar tttoiioon ^CP and SP be in tlie mm direction
about die poinu C aAd S> the point P rtiall deMibe a Ikw of the
ttfird otdef th^ has a doable pdnt in S> of the 34th UtA, ac<
eordthg to Sir ImM Newton*^ omuneMtionraiid ifCTbe
taken ut>on SC produced beyond C equal t6CS> and lihe angle
8PN be mad^ eqaal to CPT^ aeeording to the last ^Oj^ositaon^
PK AiaU be the tang^^t. If CRbetabenfrdmCeqaalloST,
biit in a 05ntmi7 dite^on^ the asya^ptete AM pM( through R ;
Md, Sd being parallel to CA, if the angle aSD be made eqnel
tOtfM on the opposite side of Smt^ the a^yiftptote shall be paral^
rd to St>. If CP meift th« right line 8D giten in position in
Mj MP shaH be atir^ys e^al to MS ; and thet^fore tliia curve
coMeMes itvth one^ of irhieh several properties are demoftstrat*
tMeiewhet^(kfmnn9,pi[tA.De8er^.turvAt.)lf^ktns^
tiM of JSP (fii^. IM) abcmt S be triple of the angnlar motion
of ^P abbnt C, and be in the same direction, so that TSP be
always triple of TCP^the point P shall describe a line of the
third ord^ ef the some specie* with the former, only in this
curve CT is triple of CR. If AM perpendicular to CS bisect
it iti A, and CP meet AM in M, SP shall be equal to SM or
CM ; becattse the angles SMP, SPM are equd, being each
double of SCP : therefore, since the angle TSP is trq>le of
TCP, the area TSP is triple of the triangle ACM ; and if ACK
be an angle of 60 degrees, the area bounded by the curve CPT
and right line CT shall be triple of the triangle CAK. The
limit of the atea included betwixt CR, the asymptote RX, and
the cnrve, is also triple of the same triangle. Let CpT be the
curve called the folia t, that has the same asymptote RX,
the same double point in C, and passes through T ; let PV
perpendicular to RT meet the ibliat in p; then PV shall be to
pV in the invariable ratio of AK to CA, or of the square root
of 3 to unit. And hence the quadrature of the foliat and its
tangents may be determined : for the area SpT must be to
Spi (or SCAM) as 1 is to •?; and, the angle CPN being
made equal to SPT, if NP meet RT in n, the right lines

1

1

I *

Chap. X. 4kd Asymptotes. Q65

fn^ jm Bhldl touch the tvltr^B TPC, TpC respectively. It will
appear afteilraitis, that the ctinre TPC is one of those which
can b^ defioribed bjr acentrioetalfbrce directed towards S that
is reeipfocdiy an the cube tf SP the distance from S. Tlie
equatitm of tl^ eurve is ttjnly 'cbmpnted from hence, that the
$ifMt ai PVisto Hit sqaarc of CVas tV is to RV (Jg. 125). If
the an^ P8C be triple of PCS, then ST is to be taken from
S towards C eqirtdtd one fetirthpart of SC; the curve describe
ed l^ P is also of the third order, having three asymptotes
that pass thr6ttgh R (CR being taken from C towards S
eqtiiil to one foorth part of OS), one of which is perpenditolar
to CS, «nd the othfir tWo form each an angle with CS that is
hiJf a right ^M. In 6ther instances, when the angle ACP is
tddSP lA a given rAtio, the tatigeifts and asymptotes are deter«
ttnMA With the same faciKty.

S17. Ex. 2. Let'therightlinesCM, SM (/g. 126)bealways
iflflecU^ fW>mtlie given points C, S to the carve BM ; and if the
pohit P be taken upon CM so/ that the triangle SMP may be al-
ivays isosceleB, lli^ tangents attd asymptotes ofFP the curve de«-
scribed by P may be determined from those of the curve BM by
the lltst proposition. Let EMbethe tangent of thfe curve BM,
and, first, let MP be always equal to MS ; make the angle SMV
eqtal to CME on the opposite of SM that CME is of CM; let
MT meet CS m V, take ST equal to 2SV, join PT, make the
angle CPN equal to SPT the contrary way from CP that SPt
is from SP, and PN shall be the tangent of the curve FP. For
the angle MSP being always equal to MPS, it follows, that the
angular velocity of SP aibout S h one half of the sum or of the
difference of the angular velocities of CM and SM about Cand
S, according as ME meets CS produced beyoiid C or S,or be-
tween C and S. Therefore, if we suppose that ST is to CT as
the angular velocity of CP is to the angular velocity of SP one
half of the sum 6f CV and SV, or of their difference, shall be
to SV as CT is to ST. Therefore ST must be equal to £SV ;
and, when CP ttnd SP become parallel, CRis to be taken equal
to 2SV, in Order to determine the asymptote RX. When M is
always in the sam^ right line, P is always in a Une of the third
order 6f the 3Sd sort, and h^ tio double point, unless that right

R 4 line

264 ' Of determining the Tangents Book I«

line pass through S; in which case the curve FPcoiiicides with
one of those which we considered ia the last artial^« But if we
suppose that (he side SP is always equal to SM^ then> the rest of
the construction being the same^ ST is to be taken, equal to SV.
Jja^tly^ if PM be always equal to PS^ ST is to.be taken equal
•to one half of SV. . Of this last case we. have au example in
the ellipse and hyperbola ; for when M {Jig. 1 27)is always found
in a ci rcle described fromtije centre Cj^ P is found in an ellipse or in
an hyperbola according as the point S is within Pt without ,tbe
circle, C and S are the two/oci, CM. is eqnal to the transverpe
ftxiSji apd the angle CME is a right one : . therefore, SMV be*
ing made a right angle, siiice SP is equal to PM, if PH bisect
SM in H, it shall bisect SV in T, and shalUouch thecufve FP;
tecausf , the^ angles SPT, CPN being equal, PN and PT coin*
cide. If, instead of a right line revolving about C,we suhsti-r ,
tute a right line moving parallel to itself, the same<:on9tru^tions
may be easily adapted for determining the tangenta apd asymp*
totes of the curve that shall be described by P (^g^ 128).
And, supposing M to be always found in aright Hne, we shaUhav^
an example of this in the parabola when MP is always equal to
PS, and in the equilateral hyperbola when MP is equal to MS.

31S, JGx. 3. Let the angular velocity of CP about C (^. 129)
be to tbc angular velocity of CM about C as any right line Cm
istoCS; and ^he angular velocity of SP about S to the angular
velocity of §M about S as Sn is to CS. Then, the rest of th^
construction being the same as formerly, let ST be to CT in the
ratio compounded of that of Cm to S;i and that of SV to CV,
This and the preceding examples Height be easily rendered more
general.

319. Ex. 4. Let the invariable angles PCG,KSH(/g. 130)
revolve about the given points and S; let M the inteisection
of the sides CD, SK describe the curve BM, and let FP be
the curve described byP the intersection of the other sides CG,
SH. LctAM be the tangentpf thecur\'eBM,maketheangleSMT
equal t6CMA,joiu TP, and make CPN equal to SPT, with the
precautions that havebeen mentioned so often,andPNshaU bethe
tangent of the curve FP. For the angular velocity of CM about
Q is to the angular velocity of SM about S as ST is to QT, by

prop.

Chap. X. and Asymptotes. C65

prop. 1 8. The angular vebcities of CP and SP about C and S
are respectively equal to theangular Yelocitiesof CM and SM
about the same pohits, because the angles MCP^MSPare inva-
riable. Therefore the angular velocity of CP is to the angular
velocity of SP as CT is to • ST ; and^ by the converse of the
l^th proposition, PN is the tangent of the curve FP. When
M comes to m, let SP and CP become parallel ; and, ma being
the tangent at m, let SmQ be made equal to Cma ; and^ if tnQ,
me^t CS in Q, take CR equal to SQ the contrary way from
C that Q is irom S, draw RX paraUel to CP or SP, and it
shall'be an asymptote of the curve FP: but if ^itQ be parallel
to CS, the branch of tiie curve described by P while M comes
to m shall not be of the byperlx^c kind. When the point M
de^bes a right liue, P describes a conic section, unless when
CP i»nd SP coincide at tlie same time with CS; in which case P
libewifi^^ describes aright line. WbenM describesa conic section
that passes through one of the poles C, Sxmly^ and the ri^t
line CP, SP coincide not with CS at the same time;P describes
a line of the third order that has a double point in that pole.
When the conic section passes through neither of the poles, P
describes a line of the fourth order that has threfe double points :
and by these constructions the tangents and asymptotes of those
curves^ and of such a9 can be reduced to similar classes of the
higher orders, are determined. While the invariable angles
}if CP,MSP revolve about C and S, and M describes a right line,
if dbe given angle P/p revolve about another given pointy^ and
the Biie/p always meet CM in j> ; then shall p describe a line
ic^.the third order that shall have a double point in C, because
JVdescribes a conic section that passes through C. The tangent
eXp is determined by drawing P/, so that the angle /P/ be
^equal to SPT, meeting C/in /, joiningp^, and constituting the
tangle C/m equal io fpt : toxj^ft shall be the tangent ; because
the angular velocity of^ is equal to that of/P^ and isto the
angular velocity of CP, or of Cp, as C^ is toy?. .

320. Ex. 5^. Let the angle MCP(yifg. 131)revoIveaboutC,asin
the precedingexample ; but, the angle PQM being invariable, let
the angular point Q describe the curve IQ, and the side QM
always pass through the pole S ; let Aa, Bb, and PN be tan-
gents

dOS^ Ofdttermimng the Tangents Book t

gents OfthecnrvesIQjGM, and FP respectively. Let the an-
gle QSD be made equal to DQa (so that a circle described
tbroiigbS,Q,andDmaytoachAa); joiaCD^andlettheaiigite
DPT, SMH be made equal to the angles CPN, CMB respec-
tivel J, with the precavtion so often mentioned ^ j^n TH, and
ii.diaii be parallel to SD : and hence the tangent of IQ be*
lug given with the tangent of atfaer OM or ¥9, the tangent of
the other is easily determined* The asymptotes of the branches
that are described by M ithen CM and SM become |>araDeI,
or by P when CP and QP become parallel, are determined by
a conatmction that easily foUows fix>m this. When the point
N describes a right tine, and either of the points M, P describe
ako a right line, the other point describes a hne of the tbhrd or*
der(scHtte cases excepted) that has a doable points The Se*
monstration of thisconstraetion iseasily dedaced from pr<^. 1 8.
Eor it follows from what was shown there, that the angular vis
locity of DQ about D is equal to the angular velocity ofSQ
about S; sO that if Aa, PN, and Bft are the tangent6> CT must
be to DT as CH is to SH, and TH parallel to SD.
S2l.Ex.6(fig. 132). Thelmesof thethirdorderthathavea^toti-
ft/epomtare alsodescribed by Pwhen the rightIinesCM,DM,and
SP revolve about the poles C, D and S, so that M describes a
conic section that pass e s through C, and SP is always parallel
to DM. Let Mt touch the conic section, and, the angle DMV
being made equal to CMt, let MV meet CD in V ; draw VH
parallel to DS meeting CS in H ; and, the angle CPT being
made equal to SPH with the usual precaution, then PT shall
be the tangent of the curve described by P. One of the asymp-
totes is parallel to CD, and its position is determined by pto-
dncing CD till it meet the conic secticm in £, drawing BQ
pmrallel toDSmeetingCS in Q,and taking CRequal toSQaccord-
iag to the last proposition. Hence a method b easily derived
fortbrawingafine of the third orderthrough six gi\'eA points,one
df which is to be a double point, that shall have an asymptote
pasallel to a right line given in position, or a branch whose
taugei^t approaches to parallelism with thisright lineasitslimit,
m those curves always have. When M describes any other curve,
the tangents and asymptotes of the curve described by P am

... - deter-

Y^aitySffBaSfiSVoll.

CS. X

/

/

T^-i»4-

I^joS.

•> ^

1

Chap.X. f and J^fn^totei. fl67

determined in the same manner. Thns, in the coneboid, if
mH (fig. 108) parallel to SA meet SH paralldto ADin H>aad
the angle QMT be made eqnal to SMH> MT is the tangent,
ses. Ex. 7, Let the right lines CM^ SQ^ DV {Jig. iss) rerolre
about the pointsC, S> and D ; let the intersection of DV with
CMdescribe thecorv^GMjandits intersection with SQdescribe
die conre HQ ; let B6 loach the former curve in M, and Aa
toach the latter ih Q. J<Mn DC^ DS^ and CS, make the angle
DML equal to CMB(widi the usual precaution)^ and let ML
meet CD in L; let the angle DOT be equal to SQA, and let QfT
jneet D8 in T ; let LT meet CS in H, join PH, and make the
angle CRN equal taSPH the contrary way from CP that SPH
is ftom SP^ and PN ^aU be the tangent of the curve described
by P. The same construction is easUy adapted to the case when
the right line MQ moves parallel. to itself, instead of revolving
about a given point. The asymptote of the branch that is de-
ecribed when CP and SP become parallel is determined fay tak-
ing CR equal to SH in that case, according to the last proiKK
sition. When M uid Q describe right Knes, P describes a conic
section, if the revolving lines CP and SP do not coincide with
CS at the satne time (in which case P describes a right fine),
as may be demonstrated by lemma £0, lib. 1, of Sir lioae
Newton's Principles, and by several other methods: and hence
a way arises of describing a conic section through five given
pdints similar to the first of those that are given in the 2Qd pixv-
position of the same book. If miote pdes beassumed in the
same manner, and the kitersections of the revolving fines that
are necessary in the description for determining P, move infixed
dg^t lines, the point P AM still describe a conic section. But if
any of them are made to describe conicsectibbsorcnrvesof any
other orders, the line described by P may arise to the order that
ii expressed by the double product of the numbers that express
t^<Me orders; and its tangents may be determined from thesame
principles. In this example, where we make use of three poles
Oftty, when the point M describes a conic secUon that passes
ihtough C, or D, and Q describes a right Une, the point P de^
s^fibes a line of the third order having a double point in C,or
S (aoiHe eases excepted, as when CP and SP coincide together

with

^268 Of dctermimng tht Tangents . Book I.

tvitb CS) : and hence other methods arise for drawing sach
curves through seven points^ one of. which is supposed to be
double, and for determining its tangents and asymfi^otes. The
conic section GM may be left out in explaining the nrnnnerin
which the cnrve is described^ or that in whidhits tangents and
asymptotes are found, by substituting the construction by which
the points and tangents of GM are determined in its place; of
which substitution* we gaVe'an example at the end of art. 31 9.
This may be easily extended to. some classes of the higher or-
ders of lines. The lines that are described in this mlinner have
always one or more of those points that are called, double, or
that are more complex : for though, in particular cases, some
of those points become simple, . yet one or more still remain
double^orthecutvebecomesaconicsection. Thus,if MC%. 134)
describe a conic section that passes neither through C nor D^
while Q describes a right Kne AQ, P describes a line of the
fourth order, that has three double p<rints, viz. C, S^ and the
point where DC meets AQ ;. unless when CP and SP coincide
at the same tim<? with CS, in which case the points C and S
are no. longer double. Mid the curve described by P is of a
lower order ; but the curve is then eitlier of the third order,
and has still a double point, or becomes a conic section; and
the lines of the third order, that have no double point, cannot
be described, when the angles or right lines are all made to
revolve about fixed points or poles, by. any method at least pub*-
lished hitherto. When the angular points jire cairied along
right lines, as in the following example^ such curves may be
comprehended in the description likewise.
323. Ex. S.The angles PML,PQL O^g. 135) bemg invariable,
let Aa,B6,andDe2bethe tangents of the curves described by the
points M , Q, and L, and let PM alway spassthrough agiven point
C, and PQ through a given point S ; make the angle MCG equal
to GMfl or A ML, ftSH equal to BQL (so that a circle through
C, M, and G may touch Aa, and a circle through S, Q, and H
may touch B6),and let CG, SHmeet LMand I^in Gand
H respectively; join GH meeting Ddin I, make the angle GLZ
equal to HLI, with the precaution we have often mentioned;
let CT be to ST as GZ is to HZ, so that the points C, T, and
S may be in^thc same situation with respect to oacb other as G,

I and

I and H ; and if the angle I^PN be made equal to Cl^ th^
contraty way from SP that OPT is ftt)m CP, then PN shall be
the tangent of the curve described by P, When M, Q, and L
describe right lines^ P describes a lineof the fonrth order^ in
which C and S are doublepoints^ unless the revolving lines CP
and SP coincide at the same time with CS. Butin this last
case (that is^ supposing CS to meet Aa «nd Bi in A and B^ the
angle SBK equal to SQL^ and BK to meet Dd in K^ when
CML is equal to CAK) P describes a line of the third oider^
the points C and S are simple (Descrip.cnrvar.prop, \5yparA),
and the curve intersects CS in a thiixi point n, the same to
which T c6mes when M conies to A and Q to B. And* tims
lines of the third order are described that have no doubk point,
and their tangents and asymptotes are determined from the
description. The same principles serve for determining the tan-
gents and asymptotes of the curves that are generated when
more lines^ angles, and poles are made use of in the description.
324. These examples may serve to show how theorems re-
lating to the description of lines are of use in resolving pro-
blems concerning them ; and this will farther appear when we
come to treat of the curvature of lines. * The tangents at the
points C and S ip the precediqg examples are determined
more easily, from art. 2 1 4, in a manner that may be applied in
some cases to any other points of the cur\'e. For example, let
A,'B, C, S (fig. 136), and E be five points in a conic section,
and let it be required to driaw a tangent at C ; join BC, SE, and
let them meet in D ; letSC meet AB in g, and ^Dmeet AEin n ;
join Cn, and it shall be the tangent of the conic section at C.
Again, let a line of the third order have a double point in C
(fig. 137) and let it be required to draw a tangent to any other
point in the curve, as S. Let any right line through S meet the
curve in other two points/*and d, and let another right line
meet the cur^^e in the three points a, b, and e ; join Cr meet-
ing Sd in D. Let a conic section described through the five
points C, a, 6, d, andy^ meet CS in g, join Dg meeting ab in ft,
then join Sk, and it shall be the tangent at S. Or the point ^
may be found without describing the conic section, by produc-
ing db and aC till they meet in /, /S, and a/' till they meet in

h, and

^9 Of the ligwts tJkt hwc Af^mptota, Book I.

k, aQd joipipg bh which product (if neceMurjr) shall meet CS
10 g. Hik tangei^t may be detennined by several other me-
thods from the same principles^ ami some properties of sack
lines may be dedooed analogous to those demonstraled cf the
conic sectkms by the learned Mr. Simson, Sect, conic. Sb. 5,
prop. A5, Sfc. Many other examples might be brought of this
kind; bnt we have insisted on this subject at a soffieiant
length. We have described these examples in this rather than
in the seventh chapter> that we might consider the tangeiits
and the asymptotes together.

Sfi5. By an assignable magnitude we always understand
any finite magnitude^ and by an assignable ratio that of any
finite quantities.

PROf^. XXVII.

Let Pr, the ordinate of the figure AVrg (fig. 138), he to PN,
the ordinate of the figure APNF, in any ratio that ap*
proaches to an assignable ratio, as its limit, while thefigures
are produced ; let PN be reciprocally as any power of BPj
cind if the exponent of this power be greater than unit,
and AP be taken upon BA, produced beyond A, the areas
AV^VyAPrg shall both have limits; but if that exponent
be not greater than unit, those areas may be produced till
they exceed any given space. It is the contrary when AP
is takenfrom A tQvcards B.

Let IM (fig. 107), in art. 29S^be reciprocally as any power of
CI (or BP) whose exponent is any positive number or frac^
tion expressed by 0i,and the fluxion of IM shaU be to thefitu*
ion of CI (or of the base AP) in the ratio ccmpounded of that
of IMtoCIandthatofflitoiuiit(art. 10? and l6g). LetPNbeto
P(j (at in art.fl93)asthefiuxjonof PM(or of IM)istotfaefittx-
iauof thebsseAP: then PMsbaH be asIM direct>]r^andBPia-
«eisely,and tberefbreinveriely aa some power of BPishoseexpo*
nentexceedsunitby m. When APia takennpon BAprodiK^dbe*
yond Ay the area APNF is always less than the rectangle cw9r

tained

Ch«qp. X. imd of the Artw HM hmt Umiitf. tft

tained by aK find PQ> which h its limits by «rt. 093* But if
AP be taken frQm A towards B, the area APNF may be pro*
duced till it exceed any given 9pace> and has no mch limits by
4ft« 39i. Therefore/ conversely, if PN be reciprocally as any
power of BP whose exponent exoeeds unit by any integer num-
ber^ or by any fraction how small soever^ ttien the axea APNF
bss a limit when AP is taken upon BA produced beyond A^ but
has BO limit when it is taken from B towards A. From this it
easily follows (and it may be likewise shown from art. 205),
that, when PN is reciprocally as any power of the base whose
exponent is less than unit, the area APNF has a limit which
it never can exceed if AP is taken from A towards B, but has
DO such limit if AP is taken upon BA produced beyond A. Thia
being premised, because the base AP (Jig. 138) is an asymptote
of the£gure APNF, and the ratio ofP»* to PN approaches to an
assignable ratio as its limit while the figure is prodnc6d,it follows,
thatAPisalsoanasymptoteofthethefigureAPrg, Supposethat
Ag is an ordinate at such distance irom B, that when AP is
taken upon BA produced beyond A, Pr never coincides vidiany
asymptote the figure APrg may have parallel to the ordinates;
and since the ratio of Pr to PN is never greater than an assign-
able ratio while AP is any assignable distimce, auid the limit to
which tfaeirratio approaches (whether by increasing ordecreas-
ing^ while the %ure is produced, is also assignable; it follows^
thai ibae may be a ratio of a greater finite quantity to a lesser,
which Biay exceed any ratio of Pr to PN. Let AA be to AF,
and the ordinate Px always to PN, insuchar^o; and the figure
APxh 01 all be always to APNF in the same ratio (by art.
1 U). Therefore, if there is a limit which the area APNF ne-
rer can amount to, there is also a limit which the ar^ APxA
never can amount to, which is to the former as AA is to AF.
But Px h always greater than Pr, and the area AFxt greater
than APrg ; therefore there is likewisefalimit which the areaAPrjf
nevercan amount to. If the area APNF (/g. 139) maybepro-
duced tin it exceed any given space, that is, if it have no limits
then let Pj: be supposed to be to PN in any invariable ratio
less than the least ratio of Pr to PN, and the area APxA shall
be always less than APrg; but since it is in, an invariable ra-
tio

27^ Of the Tigurts that have AsyntptotH, Book I.

tio to the area APNF> it follows that it has no limit. There-
fore there is likewise no limit which the area APrg may not ex-
ceed. When the figure APNF has an asymptote BV parallel
to tlie ordinates^ the figure kVjrg has the same asymptote : and
it appears in the same manner^ that the areas bounded by a
given ordinate Ag (which is supposed to be taken so near to B
that the figure APrg has no asymptote parallel to the ordinate!*
betwixt A and B), the asymptote BV, and the curves, either
have both limits, or may be both produced till they exceed any
given space. Therefore what was shown of the area APNF is
applicable to the area APrg.

326. Cor. I. It is obvious that this is agreeable to what is
shown by the celebrated author of the Geometry of infinites,
after his manner, in art. 1430. That when the base is infinitely
produced, and the ordinate becomes an infinitesimak>f any or-
der beneath the first, the area is finite; but when the ordinate'
is an infinitesimal of the first order, whether it be a complete
infinitesimal (as he expresses it) of that order or not, the art*-
18 infinite. For when PN is reciprocally as any power of BP
if the exponent of tins power exceed unit, PN becomes an in-
finitesimal of an order beneath the first, according to this
doctrine, when BP is supposed infinite; but if that exponent
be unit, or less than unit, PN is an infinitesimal of the first order.

SQJt. Cor. IL To give an example of this proposition, let
Pr be expressed by any fraction whose numerator and denomi-
nator consist of terms formed from the powers of BP and given

quantities (as by *" J , ^'wl^ere x is supposed to

represent BP, m and n are any given numbers. A, B, a, ft, SjC
invariable quantities); then, if the exponenjt of the highest
power of BP (or x) in the denominator exceed the exponent
of its highest power in the numerator, the base lihall be an
asymptote of the figure APrg; and if thisexcess he greater thaa
unit (that is, if it exceed m-^1), there shall be a limit which
the area bounded by the curve and asymptote never can amount
to ; or, to make use of the usual expression, it shallhave afinite
value .even when it is supposed to be produced in£nit6ly. Bul,-

if

Chap. X. and of the Anat that hate Limils* 973

if that excess be equal to unity or less than unit (that id, if n
be equal to 991+ !> or less), the area APrg shall have no such
Ijimit ; or, according to the usual expression, it shall become in-^
j^te when produced infinitely* If the lowest exponent of BP
(or x) in the terms of the denominator of that fraction exceed
its lowest exponent in the numerator, the figure shall have an
asymptote BY parallel to the ordinates; and when this excess
10 less than unit, there is a limit which the figure bounded by.
the curve and the asymptote BV never amounts to ; but whm
this excess is equal to unit, or greater than it, there is no such
limit: if any power of Pr be expressed by such a fraction, then
the excess we have described b to be divided by the exponent
of .this power, and the quotient in place of the excess is to be
compared with unit. For let PN be recip]:ocally as the power of
BP, whose exponent is equal to this quotient, and the ratio of
Pr to PN shall approach continually to an assignable limitwhea
the figure b produced : because a term in the value of Pr where
BP is of any dimensions, is to a term where it is of any lower
dimensions, as some power of BP is to a given quantity ; and
the latter of these terms is to be neglected in respect of the
former, when the limit of the ratio of Pr to PN is required
while the figure is produced along the base ; but the former in
respect of the latter, while it is produced alopg the aqrmptote
BV. And, therefore, the limit of the ratio of Pr to PN depends
on those terms only of the numerator and denominator of the
value of Pr where the exponents of BP are highest, in the for-
mer, and where these exponents are lowest) in the latter case.

S2a. When the ratio of the fluxion of Pr (fig. 1S8) to the
fluxion of PN approaches continually to an assignable ratio^
as its limit, while the figures APr^, APNF are produced along
the asymptote AP, the ratio of the ordinates Pr and PN to
efich other approaches likewise to ^n asaig^aUe ratio. For
Jet ly.and Pit the orcUnates of the figures A]^,APii/'mea8nre
the fluxions of Pr and PN respectively, the fliixion of the base
beiog'measiired by the giyen line D6. Let AP lie produced
to any point q, and, Q being any where upon Vq, let the o|r*
dinates «t Q and q meet the curves FN, gT,fn, and df.'vfi }IL, V,
T, Z, m, If, f , and z ; and let HT be at so great a distance from A

VOL.L S that

/

d74 Cftkc ttgurci that have Jsjfn^toUs, Book 1.

that the figures APnf, APJd may have no asymptote parallel to
the orcTinates beyond Tf. The rectangles contained by Pr and
DG^andby PNand DG^are the limits to which the areas Pjrj|^
Pqtn, continually approach while fq is produced^ by art. !^6^
and the ratio of Pr to PN is always the ratio of those limits*
The ratio of P j^^to Vqtn is always less than the greatest ratio of
the ordinate QZ to QT^ and is greats than the least ratio of
QZ to QT^ when this ratio is variable ; and it coincides with this
ratio when it is invariable : therefore the ratio of Vqtf to P^fn
alwAys approaches to an assignable ratio as its limit while Pq
is produced^ however great the distance AP may be ; aml> oon*
xsequently^ the ratio of Pr to PN approaches likewise to such a
limit. Tliis is sufficient for our purpose ; but it might be ^own,
that this limit can be no other than the limit of the ratio of
P/to Fn^ or of the fluxion of Pr to the fluxion of PN.

PROP. XXVIIL

8^. The ftux%(m of the ia$e AP (fig. 138 md 139) ttmg
given, kt the ^fluxion of the mrdivMe PR be repratmtid by
Pr, and the Ufkit of the ratio afPrtaan ordinate of an

* hyperbola that is redprocaUy as a power of BP whose e js
ponent isnbe assignable : then the figure APRa shall have
an asymptote thai is parallel to the base, or coincides with
it, when n is greater than tmit ; and there shall be a limit
which the area bounded by the curve aR and this a^^n^ote
never can amount to, if n be gftater than 2»

let Atf paralleLto the ordinates me^t the curve in a, and op
)^affalld to the bflse meet PR injp* The figure aptt haa an
asymptote parallel to the base when the area AP«^ has a limit,
by art. 29S, that is^ by the last proposition^ when the ratio of
Pr (which measures the fluxion of PR) to PN approaches tb
an assignableratio as its limit> and PN is recifMrocaUy as a power
cf BP whose exponent n exceeds unit. This a^mptote ia do-
ietmined by taking aK upon Aa fixmi a parallel xopR, and in

the

Chap. X. ^md ^fthe drua thai hav€ Limifi. 275

the nine directk^, 8o that the rectangle contained by iiK and
DG may be eqoal to the Umit of the atea APv^^ and drawing
KI parallel to thebaic^ by art, fl98. The asymptote sometimea
comcides with the baae^ ad when |»R is taken from ji towards P,
and oK is eqnal to Aa. If the exponent n be not greater than
nnit^ the figure has no asymptote paralld to die base^ by art.
298 and S25^ In the same manner it appears, that the figure
has an asymptote parallel to the ordinates when n is equal to
tmit, or greater ; because AP being taken towards B, the figure
APi^ may be produced in such oases tiU it ezoeed any given
space, by art. 325.

3^0. To demonstrate the latter part of the proposition, let
Kl be the asymptote of the figure, that is parallel to the base
when n exceeds unit (by the last article) ; let KI be taken vugh
on CK produced towards O, and let Vn measure the fluxion of
PN, which is supposed to be reciplrocally as a power of BP
whose exponent exceeds unit. Then (by art. l67),'^he fluxion
of the base being given, the fluxion of PN is as JPN directly,
and BP inversely ; and, consequently, Pn is reciprocally as a
power of BP whose exponent exceeds 2. Therefore, if Uie ra^
tio of Pr (that measures the fluxion of PR, or of IR) to a
•quantity that k reciprocally as the same power of BP approach
to an assignable ratio, as its Kmit, the ratio of P^ to Pn, or of
the fluxion of SB. to the fluxion of PN, and (by art. 328) the
ratio of IR to PN, shall approach to such a limit ; and, be^
cause there is a limit which always exceeds the area APNF, it
follows (by the 27lh proposition), that there is likewise a linrit
whidi alivays exceeds the area iiRIK. But if the exponent m
be not greater than 2> the area aRIK may be produced tiU it
exceed any given space, by art. 325, becatise the ratio of IR to
« quantity that is reciprocally as a power of BP whose expo^
nent is not greater than unit shall approach to an assignable
ratio, as its limit, in this case. It appears in the same maiiner>
that if -KI be taken towards C and AP towards B, the arei^
«RIK has a limit when n is less than 2, but has no limit whea
n is equal to S, or greater. To give an example of this pro«
position, if die ratio of the fluxion of the ordinate to tb^ flux«
ton of the base be exf^ress^ by siicha fraction as wa^ describecl

S8 * ia

*:

876 . €f the FlgHte^ thai hiM A^n^iiOes, BooSL

in artSdTi thea^ if the excess of the highest exponeiit of BP
{or x) in the denominator above its highest exponent in the nn-
smerator be greater than unit^ the figure has an asymptote that
.is parallel to the base, ox coincides with it : and if this excess
exceed 2, there is a limit wbich the area bounded by the curve
and this asymptote can never amount to ; but if the excess be
equal to 2, or less than 2, that area may be produced till it ex-
ceed any given space*

' 331. Cor.L The first part of this proposition, reduced to die
jtyleofthemethod of infinitesimals, agrees withtherule thatwas
cited in art. 299- That when the figure is supposed to bo pro-
duced infinitely, and theelementof theordinate becomes aninfi-
nite8imaI,twodegrees beneath theelementof thebase, thefiguie
has an asymptote that is either parallel to the base, or coincides
.with it. For, when the figure is supposed to be produced infi-
nitely, the ratio of Pr to PN must be supposed to coincide with
its limit ; and, since PN is reciprocally as a power of BP whose
exponent is greater than unit, and the fluxion of the ordinate
PR is to the fluxion of the base as Pr is to D6, if these flux-
ions be represented by infinitel;^ small elements of the ordinate
and base, the element of the ordinate must be to the element
of the base as a given quantity is to a quanti^ that is express-
ed by a power of the infinite line BP whose exponent exceeds
unit. This rule is chiefly of ase(as the celebrated author observes)
when the curve is not geometrical, and when AP and PR do
jiot both enter the quantities that express the relation of their
fluxions ; for if PR, or any other variable qui^tity besides the
2>ase AP and its powers, enter the vdue of the fluxion (or
of the element) of PR, r^drd must be had to the ratio of that
vadable quantity to the base AP when it is supposed infinite :
and in implying this rule, the various orders of infinites, and
infinitesimals, are best determined by the powers of the base>
and their reciprocals. The continuation of this rule, and of
that given in: art. 326, will appear fiom whtt follows.

332. Cor. IL The second part ofthis proposition may be eX'^

pressed, according to the method of infinitesimals, thus : the

figure apR has an asymptote parallel to the base, and the toea

. bounded by the cune aqd that asymptote is finite^ though infit

nitely

Chap. X. andqf the Jlreas that have LimUs, 277

nitely produced^ when, the base being supposed infinite^ the
element of theordinatepR becomes an infinitesimal of an order
three or more degrees beneath the element of the base, which
IS always supposed an infinitesimal of the first order; but that
area is infinite, when the element of the cnrdinate is an infinite-
simal of an order that is only two degrees beneath that of the
dement of the base*

333. The fluxion of the base being given (Jig. 140), let the
ratio of the quantity that measures the secondfluxionof theordi-^
nate PZ,to aquantity thatisreciprocallyasapowerofBP whose
ezponentisn, approach toanyassignableratioasits limit. Then-
the figure has an asymptote that is either oblique to the base,
or parallel to it, or ^at coincides with it, when n exceeds 2 ;
and there is a limit which the area bounded by the axcve apd
this asymptote never amounts to, when n exceeds 3. This is
demonstrated fifom art. 300,3£8, and3S0,inthesamemanneraa
theJast proposition was demonstrated from art .206, 325,and 3£8^
The asymptote is oblique to the base when the ratio of the first
fluxion of the ordinate to the fluxion of the base approaches to
an assignable ration as its limit; but it is parallel to the base, or
coincides with it, when this hmit is not assignable. If n be
equal to 2, or less, the figure has not such an asymptote; and if
SI be equal to 3, or any number betwixt 2 and 3, the area bound-
ed by the curve and asymptote may exceed any limit.

334« In order to comprehend the continuation of those theo-
rems in one view, let us call the asymptote, of the^rs^ order,
when it coincides with the base of the figure; of the second or*
der, when it is a right line parallel to the base; of the third or-
der, when it is a right line oblique to the base; of the fourth
order, when it is a common parabola that has its axis perpendi-
cular to the base ; and, in general, of the order r+i, when it
is a parabola the ordinate of which is always as a power of the
base whose exponent is r. Then let PZ be the ordinate of the
figure, and let its fluxion of any order expressed by m (that is,
its first fluxion when m is unit, its second fluxion when m is 9,
and so on) be represented by Pr, the fluxion of the base being
given. Let PN be reciprocally as any power of BP whose ex*'
poae&t is apy Aumher n; and let the ratio gf Pr .to PN ap-^

S 3 proach

278 Of (he Iiguf€S thai have AsympttUa, Book I.

proach continually to any assignable ratio, as its limit, while
BP is produced. Then, if the number n be greater than m, the
figure shall have an asymptote of the order expressed by m-f- !>
or of an order that is expressed by a lesser number. When the
excess of n above m is greater than unit, there iaa limit which
the area bounded by the curve aZ and that asymptote never
can amount to, while BP is produced; when that excess is
equal to unit, or lesa, this area may be produced titt it exceed
any given apace: and if ft be equal to «i, the figure has no
such asymptote. I^t the figure be now produced on the other
side of AF, or AP be taken from A towards Q» and when n is
equal to m, or greater, the figure has an asymptote parallel to
the ordinatea, that passes through B; when fi is equal to m^ or
^hen the excess of n above m is less than unit, there is a limit
which the area bounded by the curve and this asymptote never
QQn amount to; wheA that excesa is equal to unit, or greater,
this axea may be produced till it exceed any given spaoe: but
when n is less thanfn, the figmehas'not an asymptote thproogb
B i>arallel to the ordinatesu

S36. The first part of the fast article, reduced to the s^k of
the method of infinitesimals, shows, that when BP is supposed
to be produced infinitely, and the differenc€ of the orduiate of
the order expressed by t» becomes an infinitesimal of an order
that is aa many degrees beneath that of the element (or first i^
ference) of the base as there are units in iUn^ then the curve has
aa asymptote of the order expressed by m-f*l, or of someinfi»-
lior order ; when thai djffercnoe becomes an infinitesimal of an
order that is as muiy degrees beneath that of l^e element of
llie. ba3^ as there are units in dm-fl^ tlie aiea boundfedby (h^
curve and asymptote is finite though it be produeed in&iitely ;
but when the order of that differfnc$ is only 9m degrees be-
neath that of the element of the base, |hen the area is infinite ;
and this shgiws thecoqtinuationof the rutesgiven bythe learned
author above-mentioned. Thus, fcNr example, the base being
pipduoed infinitely, when the second d^rtnce of the ordmale
}^ of ai^ ordejr fcni or more degrees beneath that of the element
of the base, the figure has a rectilineal aaiyxnptote that is obKque
U> tihe baae^ or is parallel to it;^ or coiiicides with Ui butif ^Ma

A/.

Cbap. X. imd of the Anoi that have landts. £79

JUffercMCc be of an older that is not so many degrees beneatli
that of the element of the base^ the curve has not sach an
asymptote. If the second d^crenu of the ordinate is of an
Older five or more degrees beneath that of the element of the
base^ the area bounded by the carve and asymptote is finite
&ough produced infinitely ; but if it is not of an order that b
five degrees beneath that of the element of the base^ tli!e area
is infinite*

336 Jt is easy to see fromart. 307 aud 309(^g.l 14) when the
solids generated by hyperbolic areas revolving about their asymp*
toteshavelimits^and when they may beproduced tilldiey exceed
any given solid. If aM£ be any hyperbola, the solid generated
by AP]*f F has a limit, or not, according as AP is taken upon
BA produced beyond A, or firom A towards B^ by art. 301 ; and
since IM the ordinate of the hyperbola aME to its asymptote
KI is reciprocally as some power of BP^ its fluxion is recipro*
caUy as a power of BP whose exponent exceeds unitj andj tho
square of PN being always as this fluxion (art. 307)^ PN
must be as some power of BP whos^ exponent is greater than
|, Therefore> when PN is reciprocally as a power of BP whoo^
exponent is greater than \, the solid generated by APNF about
AP has a limits or not, according as AP is taken uponBA pro-
duced beyond A,orfiromAtowa(dsB, WhenaMEC^. 1 l4)is a
logarithmic thathas BVfor itsasymptotCj thefluxionof PM isie*
ciprocally as BP, and PN is reciprocally as the power of BP whose
exponentis|; andthesolidgeneratedbyAPNFaboutAPhasno
limit in either caae^ because the cylinder generatedbytherectan«
gle ag may exceed any given solid. When aME (fig. 115) is a
parabola that touches BV in h, the fluxion of PM is recipro«
cally as a power of BP whose exponent is less than unit, and
PN is reciprocally as W power of BP whose exponent is less
than i : and in this case,' when AP is taken from A towards
B, the solid generated by APNF has a limit which it can ne«
ver amount to> vix. the cylinder generated by the rectangle
aR about the axis ok ; but it may exceed any given solid when
AP is taken on the other side of A. When PN (fig. 1 14) is reci«
procally as a power of BP whose exponent is any fraction
greater than (j but not greater than unit^ and AP is taken

S 4 upon

C80 Of the ligura that have Jnf»q>totes, Bdok f«

ttpon BA produced from A^ there b no limit wbich the area
APNF may not exceed : but there is a limit which the loilid
generated by this area nerer can amomitto; and^ therefoie^ in
this case, when the figure is supposed to be produced infinitely,
an infinite area is said to generate afinite solid. If that expo-
n^it be still greater than ^, but less than unit, and AP be tak^n
towards B^thereis ajimit which always exceeds the area APNF,
hut there is none which always exceeds the solid generated by
it ; and, in this case, when the figure is supposed to be produc*
ed infinitely, a finite area is said to generate an infinite solid.
In the fonner case, the fluxions of the solid and area both de-
crease while AP is produced ; but the fluxion of the solid de-
creases faster, and is measured by a figure which is to tfie solid
contained by the rectangle PA, that measures the fluxion of
the area, and by the given line AF, in a ratio that, by producing
AP, becomes less than any given ratio ; and hence it is not sor*
prising that the solid should have a limit, though the area has
none* In the latter case, that ratio by producing the figures
becomes greater than any given ratio ; and it is therefore easy
to conceive, that though the area has a limit, the solid may
have none. What has been shown of the solid generated by
the area APNF (fig. 13S) is to be extended to the solid gene^
rated by the area APr^ when the ratio of Pr to PN (however
variable it may be) approaches continually to an assignable ja-
tio, as its limit, while the figures aite produced along their
asymptotes. And, in general, what was shown of areas may
be transferred, with some necessary precautions, to solids. For
example, the fluxion of the base being given, if the ratio of
the fluxion of PR to a quantity that is reciprocally as a power
of BP whose exponent is n, approach lo an assignable ratio, as
its -limit, and n exceed £, there is a limit which always exceeds
Ae solid generated by the area aRIK about the axis AP, which
is supposed to be taken upon BA produced from A.

SS7. There are several other inethods by which it may be dis-
covered when a figure has an asymptote, and of what kind it is.
When the value of the ordinate is resolved into a series that
converges the faster the greater the base is, the asymptote may
^ determined from the first terms of the series vrtien they are

suci^

Chap. X* , and qf the Areas that have Ltmits. . S81

such as remain invariable^ or continually increase^ while the
kase is prodaced. If the difference betwixt the subtangent and
the absdas ^preach continually to a finite right line, as ita
limits whik the branch of the curve is produced, and the base
(or the ordinate) may increase till it exceed any given right
line^ it is obvious that the figure must have a ri^t line for its
asymptote, that meets the base at a distance from the begin-
ning of the absciss equal to that liinit^ by art. SIS. When the
curve has a common parabola for its asymptote, the ratio of
the subtangent to the *abseiss ^proaches continually to that
of 2 to 1, when the axis of the parabola coincides with the
base ; but to that of 1 to 2, tvhen the axis is peipendicalar to.
the base : and, by observing the Umit to whidi the ratio of the
subtangent and absciss approaches, paraboUc asymptotes, of
yariouB kinds, may be discovered. When the logarithmic NF
(fig. Il6) is continued on the other side of AF, this ratio de-
creases, and may become less than any given ratio, and no
parabolic figure can be its asymptote.

338. If SP (fig. 103) perpendicular from a given point S
npon LP the tangent of the curve LB approach continually
hi position and magnitude to any assignable right line, and
SL may increase without end, it is obvious that the curve BL
ihust have a rectilineal asymptote. If ST, which is perpendi-
cular on PT the tangent of itxe curve DP (and is a third pro-
portional to SL and SP), approach in the same manner to a
finite right line, the curve DP has a right line, and BL a com-
mon parabola, for its asymptote. If SK perpendicular on TIC
approach thus to an assignable limit, the curve GT has aright
line, DP a common parabola, and BL a semicubical parabolay
|br their asymptotes. In general, when SL may be incr^nised
without end, if the ratio of SL to a right line that is as any
power of SPwHose exponent is greater than unit, approach to
an assignable ratio, as its limit, a parabola, the ordinate of
which is always as the same power of the absciss, may be the
asymptote of the curVe BL ; but if the ratio of SL to SP ap-
proach to such a limit, the figure has no rec^neal or parabolic
asymptote. In the same manner, if the ratio of the fluxion of
<be angle ASP to the flujdon of the angle ASL approach to

the

1

9

t28d 0/ tie Suffau gencraiid by ike Curve Book I.

ike ratio of any number m to a greater number u, while BL ia
produced, a paiabola whose ordinate is alwap as the power of

iIm absciss of the exponent- is the asymptote of the cvrre BL.

There are still other rules by which it may he sometimes dis-
covered when a figure has an asjrmptote and the area a limit i
but we have insisted on this subject aheady at asufficientlength.

PROP. XXIX.

SSd* The figure APNF (fig. 141) being tuppoted to revolve
ubout its asymptote AF, there is a H$nit which always exceeds,
the surface generated by the curve FN> mhen there is a limit
which always exceeds the area APNF; bui when this area

' may be produced till U exceed any gmn space, the surfacn
generated by FN has no linUt.

Let at be the tangent at N, and let xt parallel to PN meet
Nt ia t, and meet Nx parallel to the base and equal to DG
(which mensures the fluxion of the base) in x. The fluxion of
the surface generated by FN is measured by a space that is ta
l^e rectangle contained by PN and Nt in the invariable ratio
of the circumference of a circle to its radius; this rectangle is
alwaysgreater than the rectangle Px which measures thefliw-
iw of the area APNF, because N^ is always greater than Jix:
therefore the surface generated by FN is always greater thau
the area APNF; and, cansequently, if this area may be produc-
ed till it exceed any given space^ the surface FN may likewise
be produced till it exceed any given space : which is one part
of the proposition. l«t FT die tangent at F meet the asymp-
tote in T, and Ag be to AFas FT is to AT; let |>r be to PN
in the invariable ratio of Ag to AF, and rz parallel to the base
meet tx produced in 0. Because the curve is convex towards
its asymptote^ tx contimially decreases while the $gure is pro-
duced (art. 184), and the ratio of Nt to Nx decreases^ an4
|s always less than the ratio of FT to AT, or of Pr to PN; an<l
tbe rectangle contwxed by FN and Ht is less than the recu

angle

runcXKRiest.voi.i.

>

]^J3.«

^

J
1

Chap. X. ahaui its A»fmptoU. t89

angle Vz which measures the fluxion of the area APrg. There-
fore the fluxion of the surface generated by the curve FN
about AP is always to the fluxion of the area APrg^ in a less
ratio than the circumference of a circle is to its radius ; and
the ratio of that surface to the area APrg is always less than
A\% ratio. But if there is a limit which always exceeds the area
APNF; there is likewise a limit which always exceeds the aiea
APf^^ because the CNrdinates PN^ P»* are supposed to be to
each other in an invariable ratio ; and therefore there is also a
limit which always exceeds the surface generated by the curve
FN about AP. Thus it appears^ to express the proposition in
the more usual form^ that when the figure is supposed to
be produced infinitely^ and to revolve about its asymptote^ the
surface generated by the curve is finite or infinite^ according
as the area of the figure is finite or infinite; as has been oh*
served by Mr. C&tt$^ If the figure^ for example, is that of the
logarithmic* and AP be the asymptote ; or if it is an hyper«-
bola, and the ordinate PN be reciprocally as any power of
BP whose exponent exceeds unit: there is ahvaysa limitwhidi
exceeds the surface generated by the curve about the asymp-
tote AP, when AP is taken upon BA produced beyond A :
but when AP is taken from A towards B, the surface generated
by FN may exceed any given space, as might be demon-
strated from prop. 27.

540. As a curve may approach continually to a right line
while they are both produced, and never meet it, so a spiral
fine may approach continually to a certain point, and not reach
it in any number of revolutions how great soever that can b^
assigned. Let a circle be described from the centre S (fig, 142
and 114), with a radius SA equal to AF in fig. 114, let A be a
point given in the circumference of this circle ; let the arch
hr be always equal to the base AP, and Sm be taken on the
ray Sf always equal to the ordinate PN ; and, since AP is sup-
posed to be the asymptote of the figure APNF, it is manifest
that the spiral hr approaches continually to the point S, as the
curve approaches to its asymptote ; but it can never reach it
in any assignable number of revolutions, because, though the
mrdi kr (equal to AP) be never so great^ Sm(equal to PN) never
TiMushes*

PROP.

€M . Of spiral Lints. . Book L

PROP. XXX.

34f. Tkc amUnutum in art. 307 (fig- 142 aiul 114) i^MS
iuppoied, let the arch hr be always equal to AP^ and Sm
equal to PN ; and the ^ralarea SAm sAa// bt always to the
rectangle ag in the invariable ratio ofDG to Hhthc diameter
of the circle hrH.

By what was supposed in art. 307^ the square of PN is to the
square of D6 as the fluxion of PM is to the fluxion of AP, or
(if M/ parallel to the base be equal to D6, and // parallel ta
the ordinates meet the tangent M^ in as ft is to Ml, or D6;
and therefore the square of PN is equal to the rectangle con-
tained by D6 and //. The fluxion of the sjHtal area Shm is
to the fluxion of the sector Shr (by prop. 5) as the square of
Sm is to the square of SA. The fluxion of the base AP^ or of
the arch hr, being measured by DG^ the fluxion of the sectcMr
Shr is measured by one half of the rectangle contained by S&
and DG, and is to the fluxion of the rectangle a^ (which is
measured by the rectangle contained by au, or DG, and /O ^
Sh is to 2lt. Therefore the fluxion of the spiral area Shm is
to the fluxion of the rectangle ag as the square of Sm (or dT
PN)» or the rectangle contained by DO and It, js to the rect-
angle contained by Sh and 2ft ; that is^ as I>G is to oSh or
^F : and^ this ratio being invariable^ it follows^ that the spiral
area Shm is io the rectangle ag as 1)6 is to Hh, the diameter
of the circle ArH. This proposition obtains, whether AP
and KI be asymptotes of the curves FN, aM, or not ; it is suf-
flcient that the square of PN be to the square of D6 as the
fluxion of PM is to the fluxion of AP, that hr be always equal
to AP, Si» equal to PN, and that PM either increase or de-
crease continually.

342. Cor. I. Becaase the sector Shr is equal to one half of
the rectangle FP, it follows, that the spiral aiea Shm is to the
sector Shr as the solid contained by ag and DO to the solid con-
tained by FP and SA, or AF, and therefore as the cylinder ge-
nerated by ag about ax, or the solid generated by APNF about

AP,

Chap. X. - Ofqntal Unes. £80

AP, to the cylinder generated by FP about AP (fig. 14&, n. e%
Thus, when FN is a right liae^ Ar is the spiral of Areki^
mcdes, the aiea Shm is to the sector Skj as the frastum
of a cone generated by the trq>ezium APNF about the axis
AP is to the cylinder generated by the rectangle FP
about the same axis; and if Sy be the tangent of the
spiral at S^ the spiral area Swmh shfdl be to the sector Sly as one
. is to three; as was shown after Archimedes in the introduction.
343. Cor. II. When there is a limit which always eitceeds
the solid generated by the area APNF about AP^ thefe is like-
wise a limit which always exceeds the spiral area Shm geoerat*
ed by the ray Sm while it revolves about S ; and this limit is
to the rectangle ad as D6 is to Hh. But if the solid generated
by the area APNF produced may exceed any given sohd^ the
area that is generated by the ray Sm, while it revolves continii-
ally about S^ may hkewise exceed any given space. In th^ saine
manner, SA beiog equal to dK, if hr be always equal t^ ^,
. and Sm to|^M, the area generated in this case by rm, wbfiil have
. a limit, or not, according as the soUd generated by the area
. ^iMIK about the axis iip has a limit, or may be produced till it

• exceed any given solid : and in this case the circloirH is the
asymptote of the spiral. If Snt be always eqtial toPZ in fig. 140,
the spiral of Archimedes is the a$ymp|k>te of the spiral Am;
and ihis might be carried further by art. 334, 4*^.

' 344. Car, III. LetFN^C^g. 143and 1 14) beacommOn hyper-
bola, BA and BCits asymptotes, and Amshallbe theCurvethat is

• called the reciprocal or htfperbolic spiral; let the arch hb beequal
.to BA, joinS^, let a circle describ^from the centre S through m

meetS^ ino, and the archmo sha^be of an invariable magnitude
equal to hb, or BA; for bt i^ to mo as Si is to Sm, or AFto
' PN, and therefore as BP (or j^^) to BA. The spiral area SsnU
is to the rectangle ag ^i)fdr is to £AF, and therefore in this
case Smh is equal to one h«ilf of the rectangle contained by BA
and the diiference of AF and PN, or by hb and mr ; and the
limit of this area is equal to the sector Shb* Let ST perpen-
dicular to Sm meet the tangent of the spiral in T, and ST shall
be of an invariable magnitude equal to hb, or BA ; because Sm
is to ST iiTthe ratio compounded of that of tlie fluxion of Sm

to

1

«86 Ofspiml Una. BmI f.

to tlie flazioil of ir (or AP) and of the ratio of Si (or AF) to
Sm; that is, in die ratio of AF to BP, or of PN (or Sw) to
BA. It is obvious^ that a ri^t line a/ parallel to Sd, at a dis-
tance Sa from it equal to BA on the same side that A is from k,
as an asymptote of this spiral continued without the

345. Cor.IV* Let 1^ peipendicular from S on mT the
gent of the hyperbolic spird (described in the last coroOarjr)
meet itin p, and the point p shall be always found in the spi«
lal whose rays decrease pmpwtionMy while the length of die
curve xp increases unifermly, which is called by Mr. Cofte
the ewnpUeattd trMfrit. For if the aiigle Sp^ be made equal to
SaiT, p^ shall be the tang^it of the curve in which p is always
femidjby art. eil;andifS/peipendicuiarto%meetj9# inf,
the triangles pSr^ I^T shall be similar and equal, and pi the
tangent of Ihis spiral is always equal to the invariable line ST.
The flnidon of this spiral «p is to die fluxion of the ray Sp
as the invariable tangent pt (or ST) is to the ray Sp^ and thens-
Ibre Sp decreases pfop6ffioiMtl(y while op increases uniformly.
Hence the points of this spiral may be found by taking Sm, up-
^n amy ray Sr^ in the same ratio to Sf as the given arch 6Ais u>
the arch it, constituting St perpendicular on Sm equal to the
invariable line ST, or to the arch hb, j<Mning mT^ and drawing
SpperpendiculartomTO^. I44)inp. Or^ let acircleinrbe de-
scribed with the radius/A equal to ST, and letyk meet the tan-
gent fo in X, and the circle in u; letudr be perpendicular on &k
tax; and>iftheaichAjf be equal tothe excess of thetangentAz
above the arch ftif>and Sn be taken upon Sy always equiJ to Sv
(the cosine ofku), then n shall be a point in this spiral. A con*
struotkm of this spiral is given by Mr. Cota, Harmon, mentut.
p. 84. It is likewise considered by Mr. Farigwm, Mem. dk
tacad.PojfA704f(fig.l^). AH those spirals in which theratio of
the flusponof the curve to the fluxion of the ray from S^ is the
sameas the ratio of some power of the given line Sft to the same
power of that ray^ are constructed flpom this spiral by taking the
angle ftS; to frSp as unit is to Uie exponent of that power^ and
S; so that the ratio of the same powers of Sjf and S6 maybe
that<tfSptoS6; for j shall be foimd in such a spiral. '

PKOP.

Chap. X, Of $piral Lines. 237

PROP. XXXl,

S46. When there U a limit which always exceeds the area APNF
(fig. 145) while it is produced along the asymptote AD, ther€'
is likewise a limit which always exceeds (he length of the spi^
ral line hm while it approaches to S ; but tf the area APNP
produced may exceed any given space^ the spiral line hm nuiy
be continued till it exceed any given line.

Let mT touch the spiral in m, and meet ST perpendicular to
Sm iu T. The fliucion of the base AP^ or of the arch hr, being
represented by any given right line DG^ let P» the ordinate of
the figure AP^ineaaure the fluxion of PN or Sm; let Pr, thc^
ordinate of the figure APrg, be always to PN in the invariable
ratio of D6 to SH ; let the square of Pq, the ordinate of the
figure APqd, be always equal to the sum of the squares of Pr
and Pn ; and^ qx parallel to the base being made equal to DG^
complete the rectangle Px. Let a circle described from the
centre S through m meet SA in o ; and, while m proceeds in the
spiral, the fluxion of om shall be to the fluxion of hr (or of AP)^
irhich is measured by D6, as Sm (or PN) is to SA ; but Pr is
to DG as PN is to Sh : therefore the fluxion of om is repre-
sented by Pr ; and since it is to the fluxion of Sm as ST is to Sm
(by prop. 16), it follows, that ^ is to Sm as Pr is to Pit, and
mT to Sm as P jf is to Pn. Therefore the fluxion of the spiral
line hm is to the fluxion of Sm as Pj^ is to Vn, and to the flux-
ion of AP as Pj^ is to DG ; and the rectangle contained by DG
and the right line that measures the fluxion of Am is always
equal to the rectangle Px that measures the fluxion of the are^
APqd. Therefore the rectangle contained by the spiral line Am
and the given right line DG is always equal to the area APqd,
But. Pq is always less than the sum of Pn and Pr, because the
square of P; is less than the square of the sum of Fn and Pr,
by the supposition ; and,conseau€ntly, the area APqd'is always
less than the sum of the areas AVrg, AVnf The rectangle con-

tained

1

268 Of spiral Lines* Bock L

tained hj AF and DG is always greater than the area APn^
by arW 2Q6. Therefore the rectangle contained by the spiral
line km and D6 is always less than the sum of the area APrg
and the rectangle contained by AF, D6. Because Pr is to PM
in the invariable ratio of BG to SH^ the area APrg is to the
area APNF in the same ratio ; and if there is a limit which al-
ways exceeds the area APNF^ there must likewise be a limit
which always exceeds the area APrg; and there is consequent-
ly a limit'which exceeds the rectangle contained by the spiral
km and D6. Therefore there is a right line which always ex-*
ceeds the spiral km. If^ by producing the figure^ the area APNF
may exceed any given space^ the area APrg, and the rectangle
contained by Uie spiral km and DG (which is always greater
thall APrg)^ may likewise exceed any space^ and the spiral hm
may exceed any given line.

547. Cor. L Ilis theorem J expressed according to the more
nsuar fonOj isj that when the figure APNF is produced infi-
nitelyj and the spiral km, after having made an infinite number
of revolutions^ reaches the point S, the length of the spirial is
finite or infinite^ according as the area APNF is finite or infi-
nite. When it is expressed in this manner^ it is one of the pa-
radoxes of this kind that has the most mysterious appearance ;
lut there is nothing more wonderful in it^ in the manner it is
here proposed^ than that a line may be continually increasing,
and the increments acquired by it decrease in such a manner
that it shall never amount to a given bne. See art. 289*

348. Cor. II. If the ray Sm be reciprocally as any power of
the arch br whose exponent exceeds unit, or if the ratio of Sm
to a quantity that is reciprocally as such a power of Ar approach
to an assignable ratio, as its limit, while the figure is produced,

' there is a certain limit which always exceeds the length of the
spiral hm. But if Sm b^ reciprocally as a power of the arch
hr whose exponent is equal to unit, or less, the spiral Am may
be continued till it exceed any given line.

349. Cor. III. "WTien FNe is the logarithmic curve, Sm de-
creasesproporf/ona//ywhileArincreasesuniformIy,andtheflux-
ion of Sm (or PN) is to the fluxion of Ar (or AP) as Sm is to
an invariable line Sc that is equal to the subtangent of the Id*

garithmic.

<Jbap. X. • Ofthe'SumsofProgremons. €8^

garithmic. Thcflnxion of Ar is to tlie fluxion of om as S& is to
Sm ; and therefore the fluxion of Sm is to the fluxion of om as S&
is to Sc ; and> Sm being to ST in the same oratio^ the angle SmT
n mvariable. Therefore the fluxion of the spiral Am is to the
"fluxion of rm in the invariable ratio of mT to Sm ; and i(ri per-
pendicular to Sr meet Tm produced in t, the spiralline hm dhall
he equal to the right line mt; and mT is the limit to which the
spiral line produced from m continually approaches. This curve
is called the logarithmic spiral.

350. As a right line, or figure, may increase continually, anA
never amount to a given line, or area ; so there are progressions
effractions which may be continued at pleasure, and yet the
sum of the terms be always less than a certain finite number,
if the difierence betwixt their sum and this number decrease in
such a manner, that by continuing the progression it may be-
come less than any jfraction how small soever th at can be assigned,
this number is the limit of the sum of the progression, and is whait
is understood by the value of the progression when it is suppose
ed to be continued infinitely. These limits are analogous to the
limits of figures which we have been considering, andthey serve
to illustrate each crther mutually. The areas of figures cannot
be expressed in many cases but by such progressions ; and when
the limits of* figures are known, they may be sometimes applied
with advantagefor approximating to the sums of certain pro^*
gressions. Let the terms of any progression be represented by the
perpendiculars AF,BE,CK,H L,4rc. 0%. 1 46) that stand upon the
base At> at equal distances; and let PN be any ordinate of the
curve FNc that passes through the extremities of those perpendi-
culars. Suppose AP to be produced ; and according as the area
APNFhasalimilwhich it never amounts to, or may be produced
tillitexceed anygivenspace, there isalimit which the sum of the
progression never amounts to, or it may be continued till its
sum exeeed any given number. For let the rectangles FB, EC^
KH, LI, ^•c. be completed, and, the area APNT being con-
tinued over the same base, it is always less than the sum of all
those rectangles, bat greater than the sum of all the rectangles
after the first. Therefore the area APNF and the sum of those
rectangles either both have limits, or both have Qfme ; and it is
VOL.1. T obvious.

sgo Of the Sums of Progremmts. Book L

obvions^ that the same is to be said of the sam of the ordioates
AF^ BE, CK^ HLy 4-c. and of the sum of the terms of the pro-
gression that are represented by them. If the curve FNr^for
example^be the common hyperbola, b thecentre,iP the asymp-
tote ; and, AB being equal to & A if AF represent unit, the aeries
of ordinates shall represent the progression 1> i> 4-, ^> |> ^c.
which therefore may be continued till it exceed any given nam-
ber^ as the hyperbolic area may be produced till it exceed any
given space^ by art. 297. But if FNe be an hyperbola of any
higher order^ so that the ordinate PN be reciprocally as any
power of the base 6P whose exponent is greater than unit, the
area APNF, and the sum of the progression that is represented
by the series of ordinates, shall have limits. Thus there is al-
ways a Umit of the sum of the fractions that have unit for their
common numerator^ and the squares, cubes, or any other powers
of the numbers 1^ d, S, 4, 4<. whose exponents exceed unit^ for
their successive denominators.

351. And here we may apply what was shown above, ia
art. 325, 4^. concerning areas, to progressions. If the ratio of
the last term of the progression to a fraction that is reciprocally
as any power of the number represented by &P, approach to aa
assignable ratio, as its limit ; then there is a limit of the sum of
the progression, or not (or, as it is usually expressed, the value
of the progression continued infinitely is finite, or infinite), ac*
cording as the exponent of that power exceeds unit, or not. If
the terms of the progression are such as ariseby substituting suc-
cessively any numbers that increase by equal differences in place

of X in the quantity At +Bir" + S^c. ^^^^^^ j^ ^t. 327,

the progression shall always have a limit when the excess of n
above m is greater than unit, but it may be continued till the
sum of the terms exceed any given numbed when this excess is
equal to unit, or less. Thus, if the figur^te numbers * of any

order

* The filiate numl)enof the first ordti ate 1, 1, 1 , 1 , 1 , l^e. These of the seeond
Older are the lueccsiiTe lumt of those of the first order, vii. 1, 2, 3^ 4, 5, ^c, uid Com
an arithmetical procresiioa. Theie of the third order are the snccessire auaa at thoM

/

/

Chap. X. Of the Sums of ProgremanMs iQl

order be divided sQcceasivdj by the ooirespondiog figuittte num-
bers of any order that is two or more degrees higher^ the sun^
of the fractions that shall arise in this manner shall have a Umit;
but if they are divided by the figurate numbers of the next so*
perior order, the progression of the quotients may be continued
till it exceed any given number. In like manner it follows from
what was shown in art. 329, that when the terms of a progres-
sion increase,buttheirsucces8ivedifferencesdecrease^andthera»
tio of the last difference to a number that is always reciprocally
as any power of iP approaches to an assignable ratio, as its li-
mit ; then there is a limit which always exceeds the terms of
this progression, or not, according as the exponent of that power
is greater than unit, or not. If this exponent be greater than
% and the terms be subducted successively from their limit, a
liew progression shall be formed^ the sum of which shall have a
limit.

352. When the area APNF has a limit, we not only con*-
elude from this, that the sum of the progression represented byi
the ordinates has a limit ; bat when the former limit is known^

of tbe wcoD^y viz, i » 9, 6, 10, 1 5, t^«. and ate the tiiut^ar mimben. Theie of tha
fouTtli order are the tucoeniTe nimi of those of t^ third, vn. 1,4, 10, SO, 95, tScm
rind an the pyiamidal mmbeit, and so on. The fij^urate nvmhen of my oider may
likeviae be found, withoat oomputini^ thoarof the preoediqf ordeia, by takinf^ thetno*
ceatiTO prodnctiof as many of the numbeia 1, 2, 9, 4, S, Vc in their natnul ovder, aa
there aie units in the number which denominates the order of figniatei re<|uired, and
diridin^ ahrayt those products by the first product. Thus the trianfuiar numbert'
am found by difidinc the ptoducta 1X3,2X9,9X4, 4X5, SX6, t^c. eadi by tte
fiist product IXS* The pyramidala are found by dindinff the products 1x5^X99
SX9X4> 9X^X5, 4X5X6, (5^. each by 1X^X9. Ingeneial, the fii^urate numbera
of any order denoted byei kxe found by substitutiaf successively 1, 2, 9, 4, 5^

tsfc. in place of x in the general expression » - »f ^ «+3 - y+^ - ^^; ^j^„ the ^

1 . S . 9 • 4 • t^c*

fiictois in the numeiator and denominator aro supposed to be multiplied by

each other, and to be continued till the number in each be equal to that which e ipr es i e a

tfie order of the figurates vequiied diminished by unit. And when a figuiate number

«f any oider ia divided by the co n espondiny. figurate of Mty higher oader, theaunen- .

tor of the 4|uotiant is inraritble, and « is in if i df oininalni nf as many ^■rrr''*^ aa

there aie units ia the difbteooe of thoauoheis that denote thowafden.

T2 we

^

«ga ijjf tie Sums of Pwogremioni* BookL

we mxy by i% approxiinate to the TaJoe of tbe latler. Let AB
lepieMiit unit; then the sum of the recUngles FB^ EC, KH^ LI,
i^. and the sun of the ordiiiates AF^ BE, CK, HL^ Sfc. ehaH
beexpressedbytbesame number. Butbecanse thecurveis con-
rex tovrardg they msymptote^the wnof those fectangies always
^xeeedA theourvilttiealaieaoTev the same base, by thetriangfea
FEQ, CKS, KLT, LMZ, i^c. and by the ettrvilineal spaces
Ft^E, EjtK, KjsL, LyM, ^c. The sum of those triangles ap*
proadbeis to one half of the rectangle FB, as its limit ; tbere^
fore the sum of the lectangles FB, EC^ KH, LL 4r«» is greater
than the cunrilineal area AFEKLMI that is over the same base
^ded to one half of the rectangle FB, the excess being the
sum o^ the spaces FiiE, ErK, KtL, I^M> ^c. that ste bounded
by the ordies FuB, EirK, KzL, S^c. and their chords FE, EK^
KL^ ^c. Hence, when AF is a term at a great distance from
the beginning of a progression of this kind, the number that ex-
jMssed the limit of the ouivilinealaf ea APNF added to one half
()f the term APgmsneaily the Tolue of AF and the subsequent
terms BE, CK> SfC. the tpaoes Ft^E, ErK, S^t. being neglected*
But we may approximate to the value of such a progression more
accurately in the following manner, that is deduced from art. 255
9fid 9Q&, and will af^ar fully afterwards. liet the limit of the
area APNF be expressed by A, the ordinate AF by a, the first
fluxion of AF (the fluxion of the base being measured by AB,or
1init)by^9thethird,fifth^andthesub9equentfluxion9ofAF taken

akemately, and always positively, fa^49/>^« Then the sam of
theprogression represented byAF,BE,CK,HL,4-c.shaIlbe found
nearly by computing A+ ^a+Jjft— 74,5^+^^1:^+, 4^c-
Or, if AR be taken towaicls ^ equal to one half of AB, the or-
dinate at R meet the curve in V, the limit of tlie area RPN V
(or its value when RP is supposed to be produced infinitely) be
n/ow expressed by A^ the first, third, fifth, and subsequent flux«-
ionsof RVtokenalteroAtely^and al«ayBpoaitively,by h^ dyfydpc.
then the snift of the terms AF, BE, CK, HL, ^. may be found

by computing A— A*+ tt^— Trfh7/+ *<^- When
it( i^ not thelimit of theprogressiofithat isrequired, butthe sum
of any number of terms of which AF is the first, and o/'tlielast;
then we may approximate to this sum by the first series^ if we

wp-

suf poae A to xepioseDi now tbe eowHineal aiea hafB, a the
idiffeMOice of AF and «/*^ frthe diffamio« of their fimtAiuiioBBi^
<f die diffecence of tbeiF third ficudaof^ aacl so on. Or^ if froiti
A tpward» Awetakeore^ual toooft Mf of AB>iai3d 4ie<jtf-
diaate yo meet the curve ia v, vfeteaDi ilhalie tweof the teoood
aeciess, prowled A represeiit ia it the area Rr^V^ i^r^rlMent die
fdijaieffeiiceof theflflpxiooaof RV ^Adro^iii^hedifaieaoe of their
third, ihixioiMy audi so €fia. . .,

f S^a. Wheni the Umit ^ the ppogfeHioiK AS^ B£> GK> H£^
^e. is-givien^, aod tbelkuitof Ihb aiie» ie<reqinilsd>.]eithefbrfaier
be P j, and) aceording to the ficst sappoutioiiy wbere^ ex^eseei
AJRy and 6^ d^ f, ^c.> txptes^ it« firsts t&iiiA^ ftftb fluidoaa,
4rc- the limit of the area » iboodrliy eoBtf^utnig P^^^ — |iH^i
ilrri^ — TttUff— "^c. JBat^ addoMdrng to die 8CM»nd 80|ipo«-
flitioa^ theliout of the arte i$ foiiiid\b3F coaqmtiiig P-^-^
'^ ttW + vrHnc/'^'^ ^be 6«bI edapmmoai hpft^vmoLtd^
to ^ ajrea Aa/*F when P ia 8lip]^osed to nqtroaent'the sum of
the t6rm»iiroiii>AF io€ff QtK fh^Ankg. the ]aAlett)>i<»t]fliediffeniQCtt
of AF and af, h the difference of their first flukiAOs^ if.thedilt
feaence of their thiid fluxioBs j ai^ds^ cm; Alnd the eeodndsef iea
appraximates to the M^ea VSl^A wbeiiia ia' the ^fetencecKf
IIV and ray h the differeoee of thtift<finik(iiiadbti8^ d thedifcv^
«ice of their third flaxioq%. m^'W^on* .Wa refer fiie farthet
explioationr of thifi, ^ ith the demDmtjratiedaaiideicaiii^lefl^ totfa^
secood.booL We sbaU now ahojrh^wffogiessieiii&offiieuitibna
^ majy^.be foui^ ait. p^as^ns. th&tabaU ha^re issa^dblr nonilMSi
^qaal, to the limit (Of (be sun of .the temis^

3^4v A.^uesof aay au^heii€lf4|iiatiiitic8 thatcmitinoally
decrease beiqggiveii)' their AuecQsiiiifre dtfieseiicefl fOEm< a ti0#
series of terms,. thessumpf whiob iceak the beginning ig'alwayli
eq^fil to thee^oesnofithe fiisttt^iiik qfthefirst seiies above its
last term. Tfausj if A^B^CjD^E»'4^. bethe termlsi of the finrf
set^Sy it is uuioifeBt that theeHmof thediffexeneesefiA and.B^
6 and Cy C and B, D and £>< i^ Uieexcesaof A'lOxyre £. ff
the terms of the fii'st series. decrease in sucba manner that bj
continuing the progression they may become less than any quan-
tity boWstoall- soever diat can be assigned (a^r the ordin&tes'to
Aeasym^te4>eeQme'lesdihiaran3rlhtet)^ by

T3 pro.

294 * Of the Sums iff Progressions. BookT.

prodacSilg the figure)^ then the first term of the first series is the
limit of the sum of the second series. In like manner^ the dif-
ferences of the alternate terms of the first series^ as of A and C,
S and Dy C and £^ SfC. form a new progresnon of terms, the
ram of any number of which is equal to the excess of the som
of A and B, the first and second term of the first series, above
the sum of the last term, and lastbut one; and the sum of A and
B is the limit of the sum of the new series. In general, if a pro-
gression is formed by taking the differences of the first term A
and the term whose place in the series is expressed by any
number n, of the second term B, and that whose place is n + 1,
<of the third tenn C, and that whose place is n+2, and so on ;
then the limit of the sum of this new progression shall be equal
tothe sum of the terms A,B,C,D, 4^. which precede that tentf
whose place is expressed by fi. In thismannerprogressionsmay
be fomid, at pleasure, that may be continued without end, and
have giinen numbers for the limits of their sums t and this
method differs not materially from that of the celebrated Mr;
Jdmes Bernoulli. ^ *

955* for example^ let the first series be 1, f, f-, i, f, ^.
The successive differences of those leims are-|,t,iV>iAr>iArj 4^«
and the limit of the smn of this progression is unit, by the last
article. If we multiply each tenn of this last series by £
(that the first term may be unit), we shall have 1, ^, |, iV> 4*c.
which have the triangular tiumbers for their successive denomi-
nators, unit being their common numerator ; and therefore the
limit of the sum of this progression is 2. He successive dif-
ferences of the terras of thjs latter series being each multiplied
by I (that the first term of the new series may be unit), give
1, i^'^y'^f SfC. which have the pyramidal numbers for their
uccessive denominators } and the limit of the sum of this
progression is4* In the same manner, thelimit of the sum of the
fractions that have unit for their common numerator, and the
figurate numbers of any order denoted by m for their successive

denominators. Is found tp be ^^* . .

^^6, The ^ame series 1, i,>^ |^f, 4*^ being assumed i^n,
th^ (liferenc?s of the ^dternate temi« are.f^ f, -f^, ^ 4rc. the

limit

Chap. X. Of the Sums of Prognmam. * figs ^

limit of the som of which progresflioa b 1 1> hy art. 354^ and^
dividing each term hy 2, the limit of the som of || |f ^ Sfc.
is i. If we take the differences of the first term and that whose
place is m, the second term and that whose place 19 iR+ \, and
so on ; the common numerator of thosd differences shall be
m — 1^ and their saccessive denominators shall be the products
of 1 and m, 2 and m^l, 3 and m-f-8, and so oh; The hmit
of the sum of this progression is the' sum of as many terms 1^ ^
^, i, i^c. as there are units in m — 1 , by art. $54, and if each
term of that progression be divided by m — 1 , that miit may be-
come the common numerator, die terms-, ^— TTf 5— To.rrT^^

^c. will arise (whe^e the factors in. the denominators that are
separated by a point are supposed to be multiplied by each other)^
and the limit of the sum of this progression is equal to the sum
of the fractions l^i^hh 4rc« (continued till thehr number be
m — 1) divided by m — 1. If we now assume the alternate
terms only of the first series beginning with the second, that is^
h h h h ^^' ^^^ f^^^^ ^ ^.^^ series by taking the difference
of the first of these i and that whose denomiBator isoi-|-l (m
being any odd number), of ^ and that whose denominator is
fit-|-d, and so on ; the new terms shall have m — 1 for their
common numerator, and the products of 2 and m4-l/4 and
m^S, 6 and m^^S, ^c> for their successive denominators: and
the limit of the sum of those fractions shall be equal to the

sum of the fractions ^ i, f, i^c. that precede ^^ y the num-
ber of which fractions i s*^ By assuming the other alter-
nate terms of the first series, that is, 1, f, \, f 1 4*c. and taking
the differences of 1 and the term whose denominator is m, of
^ and that whose denominator is m-f-1, and so on, the terms
of the new series shall have m^-l for their common numera-
tor, and the products of 1 and m, 3 and m+2, 5 and m-t4^
4^c, for their successive denominators ; and the sum of the terms
shall continually approximate to the sum of the fractions 1, ^,

i> h 4*^* ^^^ precede -• We may likewise assume any other

T 4 equi-

1

£§& Qfti^ Sum$' of Pr4»gr<s$iottti Book I.,

equidistant tcrn^' of the fiist series^ aad by takiqg: their diff^
enoe$f &mi new progressions^ the valueof whivh^shftUbe thefiist.
term tba nsi^ assumed. If> we aapume the t^ms l^ \, ^, fy, Sfc.
pUssing always eves three tenix% apd divide tbesucc^ive diffeiv
eacesofthaa^itermaby 96i the series wil}: arise that is^vea hg
the celebraHod Mr. Mtmmortj in Philos. Trajisact. n. 363, and,
i$ mturbed C ; and therefore the value of this, series (when it 10^
siq)poaed to hft eointmued infinitely), is ^, by ai^t. 354. If we
aaswne thealteroete terms of the first series deduced lAarL 355^
«j^» i> ^.A> ^ 4r<?- and divide the successive difl^rencesoS
those terms by 2, the series will arise that is marked B in the
same Transaction ; the value of which is therefore 4. If we
assume the first, fourth, seventh terms, ^c. of the same series*
from art. d55,.pa8sing always over two terms-, their differences
divided by 54 coincide with the series marked A in the same
place ; and therefore At vahxe of this series is 7^^. If we'
multiply tixe eorrespondii^g tenns of any two progressionr
hy each other, aaid if the prodticts may become less than any-
given number by continuing them (as;; for example, if we muU
tlply the succeswve terms of the series i; ^^ f, f; i, S^. by the'

successive powers of any fraction -j^ their differences shall al«

waysgivegorogresAi^a of this kind. It is obrious, from ait. 354ji
how the sum of any given number of terms is^fouiid ia the pza*
{poetsionawe have nienUoned,orin any others that are deducedi
in this manner. But what follows seemato have a nearer rela-
tion to the method of fluxions, and to be of greater use.

357- Let us assume the series -^ ;jr];, j^, ^ji^, ^c.where

unit is the common numerator, and the denominators increase
bv the continual addition of unit. The successive differences of

those tenns are — rTt \% , q i ; q . . 1 ' a . ^ >

4^. and the limit of the sum of this series <Qr its value when it

is supposed to be infinitely produced) is -^ tJie first term iA the

series which we assumed^ by art, 354. The successive differencea

Ch»p. X. 0/the l^t$mof Profffeamm^ <0^

of the tenn» of this last aeries divided, by 2 are — ^rr—rs*

Bixa^iB therefore STTUPt^' ^^ *^ ^"*^ artide. The
s^jsoeasiye difTerenices of those t^enn^ dl\;ided. eai^h by 3 aie-

^.m+l.l^2.m+^ ^+i.^2.p4-^.mJ^ *^- ™d the li-
mit of their sum is therefore 3^". ' ^ 1 1 '' wi g ' Thetermsineacfc
of those progressions- we formed from the first term hy substt<*
tuting successively in its deiioii]inatorm-(-l^m4-€^i7i4^S^«»-|-4,
4c. ia^glpce o£ ^wv And it appoaj^tbaty.in.gQueral, if we sntn
6|itate successively in^ m^l, m^Q,m^3, S^c. in place of fli

in the firaction ^ mji . wig ! tjL ' (^^^ *^^ factor in the.
qenominator are supposed to be continued till their number bet
n4"l)j A^ terms that shall be produced in this manner shallr
fbrm aprog^ession^ the yalue of which is found by substituting. 11^
in the denoininator of that fraction^ in placeof the last and greaU
est factor. This useful theorem has been demonstrated by ae^
veral eminent Hathematioians^ Dr. Taylor, Mr. Nicole, and;
of late by Mr. Stirling/^ who has much improved the me^.
thod^ of approximating to the values of progressions that arise
in the resolution of problems. It is obvious from art. 354 how
the ^Mpi of any ^yen number of, terms may be found in the
professions we l^ave described.

358- The samewif S5^ ^j, jig, — , ... -^, ^

bmig^Bsumed^ where^is^smpposed to represent aay quantity at
pleasure^ aoad n any' integer number ; let a new progression be

formed by taking the differences of the first term -- and of—]— >

of the second ^;qrj wid ^. ; p of the third ^jn^ and

^ nesmmmtumeMrieiuxD^ prop* 9*

1

1

figt Of the Sums of Progremam. ' Book I.

^. , j > and so on. Because the difference of the denomi*
nators of those fractions is always n, it is manifest that the
tenns of the new progression shall be — --r^> ^^n" min Tl^

m\i,m\n T?^ ^^* The valuc of this progression is found by
•omming op ail the terms in thefix;^t series that precedethe term

jr;> by art. SS4, the number of which terms is n. Hence

it appears;, that when n is any int^er number, if we substitute
successively m, m-f-li vi+2, m+3, ^c. in place of m in the

fraction ^ ^ . > the value of the progression that shall be thus

111
formed is equal. to ;;^+;; — — ry + " — jr^+^ ifc. those

terms being continued till their number be n. But when n is a
fraction, the value of the progression that b formed in the same
manner cannot be assigned accurately in numbers, but we may*
approximate to it readily by a method explained below (art.
361). Several eminent Mathematicians have treated of this
sobject,' besides those already mentioned,, as Mr«. Leibnitz,
Mobs. BentoulH, and Mr. de Moivre; and various me-
fhods have been given by which an infinite variety of such pro-
gressions may be found* The following perhaps may be worth
mentioning.

SoQ. Therightline AB(/g.l47)beinggiven,letthe point P set
out from^ A towards B, and proceed always in the same direction
J with an uniform motion ; let the point p set out at the same
time from B towards A.with a velocity greater than that of A ;
and let it be constantly reflected wiUi an unifcvm motion be-
twixt P and the fixed point B, so as to describe BD while P de-
scribes AD, and to describe DB -f- BE, EB + BF, FB + BG,
Sfc, in the same times that P describes I)£,£F,F6, ^c* respec-
tively: then the spaces described by P and p betwixt the
terms when they meet each other shall form two geometrical
progressions, and the conunon ratio of the terms in both shall
be that of the sum of the velocities of P and p to their diffid-
ence.

Chap. X. Of the Sum ofProgresAm. 299

ence. If we suppose the velocity of jp to increase or decrease
every time it comes to B (but still so as to be greater than the
constant velocity of P)^ and to continue uniform till it return
again to B, the spaces described by P betwixt the terms wl '
it meets p shall fbrm progressions of various kinds, that are
aily determined when the rule is given according to which the
velocity of /I increases or decreases. But to obtain progiessioiiii
whose terms may have more simple expressions^ let us conceive
p not to be reflected from P to B, but every time it meets P to
be instantly brought back to B^ and to set out anew from B to-
wards A till it meet P again, so as to describe BD, BE, BF, B6,
4«. in the same timea^ respectively, that P describes AD, DE^
EF, FG, 4^. Then the spaces described by P shall form a pior
gression of terms that may be continued without end, when the
velocities of P and p (however variable they may bc^) are al«
ways in an assignable ral^ to each other white P describes AB |
the sum of those tenna is always less than AB, but approachei
to it as its limit while die progression is continued. Suppose
the motion of P to he invariable, and the motion ofp to be
uniform while P descxibes any one term; let the constant velo^
city of P be expressed by in ; and the velocities ofp, while P
describes any two successive terms EF, F6, be expressed by V
and u, respectively : then shall F6 be to EF as V is to m -f-ti^
because FG is to BF as m is to m+tt, and BF is to EF as V is,
to m» The sum of any number of terms described by P, as
AG, is Sound by subducting from AB a right line that is to the
last term FG as u is to i^,* for such aUne is equal to BG.

360. The velocity of P being expressed by m (as in the last
article), let the successive velocities ofp increase equally, and
be expressed by n + 1^ 't + ^> ^ + 3^ ^ + 4, 4*^. and let r be
equal to m-^n. Then, since AD the first term described by P
is to AB as m is to r 4- 1^ if AB represent unit, AD shall beex«

pressed by —•* The ratio of Y to m'\'U is successively that

of ii"|» 1 to r+2, n+2 to r+S, n+3 to r-f 4, ^c. There-
fore, supposing (after Sir Isaac Newton's manner) A to ex-
press the first term —^ B the second term, C the third, and

so

80oa,tiietenB3deseri6eclbv{>bre-^ ^ • *+' ^ • '^+* c . »f3

^. Thft limit of the sum of this progieasioji is AB, or ttoit -
and the smn of any number of terms denoted by z, the last
t^bw8/,i«.l-i^. FwewuafJ,,iiwe«^po«ri^
ftst velocity of ;» to be eqnd Ky tfie constimt Telocity of P
«! Ae >«ce«m. Tefocities of|, to be f, %, », 4, 4^c. that is'
itm fte tinrt, and n ranish, t&e terms describerf by p shaH be

rrSl a7$» »?♦» m»*<^' *^® ^™» Pfwhichmust be always less
t|i«icuMt,«,i.«t.«55. AByt«nnEGwfcM»^hiceinthe«.

'^"*""*^^'»rrr5:Ti «»**l»«efere AG, the sutoof
as many t^rms from the beginning as thpre are nn^ts ia ^ i,
(by wbatwas Shawn, iu the last«tide) ; Jj,, and the wm'of
thesamemmiWof fa«rtwiw,thai»a^e«ai«fe,.tlw«^,,^^
«««»«itoT> and *« tiiMgdaif nwbte 1, 3, ^ 1<»1 4,;. ^
BucceBave denominators, is SA®, er ^, The ^st remain-

wfeif«be«qiMltoa,thBttem»de8crite4l>yP«haU be — 1-
rrO*p^>*''- FGwexBressedby— -J«_g, AG bjj

* ^ x+i.x+8 5 *«<• the sum of the same munber of fractions
that have unit for their common nmnerator, and the Ryramidal

numben foE their successirc denpmioatoK i&- 3

In the same maaHer it will appear -that the suc^ssive Vtfocitt«

of ;,ben^ represented by 1, 2, 3, 4, 4:«. if the constant veloci-
ty of P be expressed by any positive integer number «, the

sum ofthe same number of fractions that have unit for their
common numerator, and the figurate numbers of the order
«+3 for tl«.r successive denominators, shall be to AG as
»+l is to m, and i» therefore equal to i multiplied by tho

excess

Chap. X. Of the Sums of Progreidoni. 501

excess of « + 1 abore the fraction ,^ :^^j .^^ir^ ' &g ^^^^

the f«ctofs aiesuppoied to be eonftjnued in the niimerator and
denomioator till the number im each be m; as has been demon-
strated by Mr. Bemoulti, deteriebus infinith, § 18 and I9.
Sfil. Supposing «ii still to vanish^ or ^ to be equal U^r, the

ienns described by P «e ;^, ^^ ^^^^, ^^^ ^^^^^ ^^
9 3 fli

is unit, it follov^s (dividing by mm) that — is the limit of

the sum. of ^;r7^f <i.H*i*H-s' ^ . H-t > H>8 > ^-f ^f
, ; ^rs^ — ,,. , V , Jjrc. From which, and what was

shown in art. 357, it follows^ that we approxiAate t^ the sun of
the terms which arise when we substitute socoessively tn,in^ I,

m^2, m^S, S^e, in place of m in the fradtioA *^ ^ by summing

uptheseri^i^5~pY» ^.m^i.^^i ^ ^^.m^i.'J^d^m^ P
*^- «' I I^x^ 3T^sf 4T^> *^- ^^'^^^ A rei>resents
the firftt term *^. Bihe second term, and m cm. Of tfaeuse of

rtiis^ see Mr. Stirling^ treatide, de summatione seritrum, p. 28,
In gelieral, when n does UQt vani^, we fbund that unit is the

limit of the sum of the progression ;^, '*^ , '^
——-y 6fc. Thereibn*, dividing by m and r, —^ is the va-

lue of the progression -7 , r-J- — -^- r^^ — ~ — -.-

4-c. And, by art. 357, we shall approximate to the sum of the
therms that arise by substituting successively r, r+ 1, ♦'+2, 4-c.

in place of r ia the fraction or ifwesumupthe

temLi

1

9M Of the Swm ofProgrcmiom. Book L

^^^^l^ST^fi'sTTW *f- ^ *e series | + r7^I

^ s/ivfg + 4, V^3 +> 4rc. When ni» any negathrefiractkm

equal to — f, ^'^';zii becomes ^^ y ) and the series becomes

I . A. I— / , 2B. 2-/ , 3C.S—/ , . ... - ^

7 + 5TT+I + TiT+T "♦" 4.r4-3 +* *^- ^°ich, therefore,

iqpproximates continually tothesamof the tenns that arise when

r, r + 1, r + 2, 4'<?- are substituted successively for r in ;r7p;

and this agrees with what is shown in a difierent manner in the
•ame excellent treatise^ p. 1£ and 26^ where this series is applied

fcr finding readily the Yalue of the progression |-^ + 5-^

+ f^ +> t^c. that was invented by Lord Brounker for

ihe quadrature of the hyperbola,

. 362. Let the point P(^g. 148) move now in the right line BA
produced beyond A^ and the velocity of p be always greater than
that of Py that it may overtake it ; let the motion of P be still
«niform> and its velodiy be represented by m ; and let the suc-
cessive velocities of p by which it describ^ any iS'wo tetms BF^
BG (while P describes EF and FG) be V and u, respectively.
Then shall FG be to £F as V is to if — m, because FG is to
BF as m is to u — m, and BF to EF as V is to m. The sum
of any spaces AD, DE, EF, FG, described by P, added to BA
(which is supposed to represent unit), is BG, which is to FG
the last of those spaces as u is to m ; and if FG the last tenn of
the progression be expressed by /, then BG (the sum of the tenns

including AB) shall be -• When the motion of p is likewise

uniform, the spaces described by P continually increase in ageo-
metrical progression ; the common ratio of the terms is that of
the velocity of jo to its excess above the velocity of P; AG the
sum of any number of terms described by P is found by sub>
ducting AB (or unit) from BG, which is to the last term F6,
as the velocity of ji is to the velocity of P, or as the number

that

Chap. X. Of the Swns of Progremtnu. sos

that expresses the common ratio of the terms is to the same nmn*
her diminished by unit : and hence the common rule for find-
ing the sum of any number of terms in a geometrical progressicm
may be deduced. When the constant velocity of P is denoted
by «i, and the successive velocities of p by r-jj-l, r+2,r+3,
4^. let 91 be equal.to r — m, and the terms described by P shall

be—^, ^1 g , 'J ^ y iiC. where A represents the first term^

B the second^ and so on. And, if we prefix A6, or unit, to
those terms, the sum of any number from the beginning to
the term whose place is denoted by z shall be to this term aj
xJ^r — 1 is to m. When m is an integer positive number,
and t is equal to m (orn vanishes), these terms are the figurale
numbers of the order denoted by m, the second number of each
order (or the first space described by P) being m. In this case,
when m is unit, each term is equal to AB, or uuit : when m
b 2, the terms are in arithmetical progression, the last term is

Z, and the sum of the terms '" - ^^ ; when m is 3, the terms
are the triangular numbers, the last term is *'^t ^ and the

sumof thetermsis '* g 3 . And, in general, the sum

€>f the figurate numbers of any order m from the first, or unit,
to the last (whose place is supposed to be denoted by jk), is

found by multiplying this last by -i— - , or by computing

" ** 1 2*3 4*t^' '^' > w^^^ *^^ factors are supposed to be
continued in the numerator and denominatortill there be as ma-
ny in each as there are units in m. We might suppose the ve-
locity of p to observe other rules, or the velocity of P to vary
likewise ; but, not to insist iurtber on this subject here, we
shall only add, that there are other methods besides that describ-
ed in art. 350 and 351, by which it may be known when a
progression of fractions has a limit, or not : such is the rule
/given by an author we have often mentioned, that when A, B,
C are any successive terms of the progression at a sufficient dis-
tance

^

304 OftkeCunfaiurexflmeu • &odkt.

laace fitMn the begtiming, and therlitio of A toC iib le«s dim
tiie ratio of the difference of A fmd 6 tb 1;he difiere&ce off B nid
C, the progression may be continaed idU its sum ^ceed any
given number ; but when it is erthemrise^ the sum of the Re-
gression ^as a limits

CHAP. XI.

Of the Curvature of Lines, its Variation, and ike different
Kinds of Contact; ofihe Curt/e and Circle -cfCurvctture, tke
Caustics by Reflexion and Refraction, the centripetal Forces,
and other Problems that have a dependence t^pon the Cur^
vature of Lines.

363. Any two right lines applied upon each other perfectly
coincide ; and the rectitude of lines admits of no variety.
Arches of equal circles applied upon each other perfeotly
coincide likewise ; and the curvatuie is uniform in all the
parts of the same, or of equal circles. Arches of tmequal cird^
cannot be appUed upon each other so as to coincide ; but whea
they touch each other, the arch of the greater circle is kn in--
fleeted from the common tangent, and passesbetwixt itand the
arch of the lesser circle through the angle of contact formed by
them, and is therefore less curve. Any two arches of curve
lines touch each other when the same right line is the tangent
of both at the samie point; but when they are applied upon each
other in this manner, they never perfectly coincide^ unless they
are similar arches of equal and similar figures : and the curva-
ture of Unes admitsof indefinite variety. Because the curvature
is uniform in a given circle, and may be varied at pleasure ii^
them by enlarging or diminishing their diameters, the flexure
of curvature of circles serves for measuring that of other lines*
There is but one right line that can be the tangent of a give^
arch of a curve at the same point ; but circlespf an indefinite
variety touch it there; and these have various degrees of more
and less intimate cofltact with it,

364. As

Chap. XL . end its Variation. -^ 305

364. As of all the right lines that can be drawn^ through a
given point in the arch of a curve^ that is the tangent which
touches the arch so closely that no right line can be drawn bei
tween them (art. 181)^ so of all the circles that touch a curve
in any given point, that is said to have the same curvature with
it, which touches it soclosely thatno circle can be drawn through
the point of contact between them^ all other circles passing ei-
ther within or without them both. This circle b called the
circle of curvature, its centre the centre of curvature, and its se»
midiameier. the ray of curvature, belonging to the point of con-
tact. The arch of dus circle cannot coincide with the arch of
the curve^ but it is sufficient to denote it the circle of curvature
that no other circle can pass between them ; as the tangent of
the arch of a curve cannot coincide with il, but is applied to it
•o that no right line can be drawn between them. As in all fi-
gures^ rectilineal ones excepted^ the position of the tangent is
continually varying ; so the curvature is continually varying in
aO cundlineal figures, the circle only excepted. As the curve
is separated from its tangent by its flexure or curvature^ so it ift
separated from the circle of curvature in consequence of the in-
crease or decrease of its curvature : and as' its curvature is the
greater or less according as it is more or less inflected from the
tangent, so the variation of curvature is the greater or less ac-
cording as it is more or less separated from the circle of curva«
ture. It is manifest that there is but one circle of curvature
bdongingto an arch of a curve at the same point; for if there
were twosuch circles, any cirdesdescribedbetween these through
that point would pass between the curve and circle of curvature,
against the supposition.

S65. When any two. curve lines touch each other i% such a
manner that no circle can pass between them, they must have
the same curvature, by the last article; for the circle that touches
the one so closely that no circle can pass between them, mu^t
touch the other in the same manner. It^nll appear from the
following proposition, that circles may touch curve lines in this
manner, that there may be indefinite de^ee» of mcnre or less
intimate contact between the curve and the ci^rcle of curvature^
and that a conic section may be described that shall have the
VOL. L U same

iM Of the Cim!tLt\&e 6f tim, )3o6kt%

^itit fsbln^toxt Tridi a gi^e^ lih^ at a giveh ^mt, ahd the ralkie
f atiafloh bf curv^ture^ bt a eonUct of the same Und ii^h the

PROP. XXXH.

806. £tf on^ oKTve EMH (fi^. \49und 150) aiui m aarck ERB
• ^^iicft tic ngfik Ale ET M €ie «0Mie 4Jd^ a/ E ; te
iinB TK'ptrdUil to the ^hard £B tifeet the tmgmt in %
SMH III ]M[> Md o ciif^e BKF MatfNism through B tii K.
ISm, $r the rectUMgh MTK d« aiK«jr» efuul t» the nqmre
^ET, thecwvatttre ^fBMHmt EnhmU be the aame^M
that if the 4it^ck BRB^ md the emtaet ofEU ««d £R
AaU he alnm^ the cbterthe Mi the engkeBthalis eontauh
' ^ri«eB%/i«oiiroeBKFalKltAem-«&^imwfttrcBQ&

list tbe nghtline TK (Jig. 14») meet the circle in R and in

<^a»d, Binoe the tectangle RTQ h equal to the sqnare of ET^ it

«UBt be equal to the rectangle MTK, by ihe supposition; wai

theiefereRTistoMTasTKistoTQ. Suppose first Aat BK

tte part ai the cnrreBKF thai is next to the point B^ ad^oiifr-

ii^ to it, AkHb without the drcle BQ ; and snfqpose the r%ht line

•TK by movii^ parallel to itself to i^proach to EB tfll it eom-

cide tridi it ; and while the point K descaibes KB^ TK betof

gtMter than TQ^ RT mnst be greater thah MT, and the arcb

EM of the barte mnst paas widioat the citde BR betwixt k

and the tangent JST: and sinds any bircfe described through

E, upon a chord less than £B touching £T> falls within the

iHTcle BRB> it is manifest that too such drcfe can pass betwixt

tile Mi^e EM and circle BRB« Let any circle Brb, described

tfpeitk a «hord Bfr greater than £B, touch ET^ and meet TK in

TtAd'if^, andj since the rectangle rTq is equal to the square of

Sr, ^ Ale veetengle MTK, MT is to rT as T; is to TK:

nitdiSiMis the-curVe -FKB passes threogh B (by the supposition)

' ao thut thiS'part of it ikat is ikelitt adjoining to B ni»st be within

' thie«t^ 2p9 of the circle Ai^; it fbUbws.that^Hiite K dasoibes

' ■i_i.

Chap. Xt. anfi %i$ Famiion. 8^7

tills part of fKB, Tj muat be greater than TK, and, coxwc*
^uentlj, MT gceater tlien rT. Therefore the arch Er of tbq
eircfe Erb is witboat the curve EM, and passes betwixt it an4
the tangeot ET. Therefore no circle whatever can pa38 betwixt
£M the arch of the curve and ER the arch of the circle; and>
cQtnequentJy, the circle ERB has the same curvature with EM
at £. Suppose now that the part of the cwve PKF (Jig. 1^0)
Ihat is next adjoining to B fall» within BQ the arch of the circle
BQE ; then, while K describes tlm part of the curve FKB>
TK being less than TQ, RT must be less than MT, and the
arch EM of the curve must faU within ER the arch of the
circle; and since any circle described through E upon a i^oid
greater than £B falls without the circle ER, it is manifest that
no such circle can pass betwixt ER and EM* Nor can any
circle Erb, described upon a chord £& less than EB, toacfaing
ET, pass between ER and EM : for letTK meet this drdb
in r and q, and MT being to rT asTj isto TK, and Tq be-
ing less than TK while K describes the part of the curve FKB
tiiat is next adjoining to B,MTmust befess than rT; and, con-*
sequently, the arch £r of the circle Eri must fall within EM
the arch of the curve. Therefore, in eithercase, all the civclei
that can be described through Brhll without both ER ^ d; £M^
or vnthin them both, and no circle whatever can passbetweea
them, widen the rectai^le MTK is ahrays equ^ to the square
of ET, and the curve in which K is always found passes through
B ; that is, the circle ERB and the curve EM have the same
curvature at E, by art^ 364, which was the first part of the
proposition.

3d7. Let Em (Jig. 151) any other curve touching ET in E>
undfkB another curve passing through B, meet TK in m andk,
and let the rectangle mTk be likewiseequal to the squareof ET ;
then the curvature of Eon at £ shall be the same as that of the
circle ERB, or that of the curve EM, by what has be^i de-
monstrated. BecauM therectanglesniTA,MTK, RT^areequal
to each other, Urn is toTMasTK is to TAr, andT«itoTRa!l
TQ is to Tk. Therefore, if the arch Bk pass between BK,
the aroh of the curve BKP, and BQ the arch dPthe circle BQ&>
the cucve Em must pass between EM, the arcb of the a^^

U 2 EMH

/

n

308 Of the Curvature of Utu$, Book I.

EMH^ and ER the arch of the circle of curvature ERB ; so
that Em must have a closer contact with this circle than EM has
with it : and the less the angle is that is formed by the curve
FKB and the circle of curvature EQB at B^ the closer is the
contact at E of the curve EMH and the circle of curvature
ERB. TIius tlie curve BKF by. its intersection with EB deter-
mines the curvature of £M^ and by the angle in which it cuts
the circle of curvature it determines the degree of contact of
EM and that circle^ the angle BET and right line £T being
given.

368. Cor. I. It appears from thedemonstraiion^ that accord-
ing as the arch BK of the curve BKF falls without, or within,
the arch BQ of the circle BQE, the arch EM of the curve
EMH falb without, or within, the circle ERB; that when the
curve FKB cuts the circle ERB in B, the curve HME cuts the
circle of curvature in E ; that when the curve FKB is on the
same side of the circle BQE on both sides of B, the curve
HME continued on both sides of £ is on the same side of the
drcle of curvature ; and that the contact of the curve EMH
and the circleof curvature is closest when the curve BK touches
the arch BQ in B, the angle BET being given, but it is farthest
from this, or is most open^ when BK touches the right line EB
inB.

369* Cor, n. There may be indefinite degrees of more and
more intimate contact between a circle ERB and acurve EMH.
The first degree is, when the same right line touches them both
in the same point; and a contact of this sort may take place
betwixt any circle and any arch of any curve. The second is,
when thecurve EMH and circle ERB have the same curvature,
and the tangentsof thecurve BKFapd circle BQEintersecteach
other at B in any assignable angle. The contact of the curve
EM and circle of curvature ER at E is of the third degree, or
order, and their osculation is of the second, when the curve
BKF touches th^ circle BQE at B, but so as not to have the
same curvature with it. The contact is of the fourth degree,
or order, and their osculation of the third, when the cur\e
BKF hfls die same curvature with the circle BQE at B, but
so as thaf fivM contact is only of the second degise ;, and this

grada*

i

Chap. XL end its Variaitim* 309

gradation of more and more intimate contact^ or of approxi-
mation towards coincidence, may be continued indefinitely, the
contact of EM and EK at E being always of an order two
degrees closer than that of BK and BQ at B. There is how-
ever an indefinite variety comprehended under each order.
Thus, when EM and ER have the same curvature, the angle
formed by the tangents of BK and BQ admits of indefinite
variety, and the contact of EM and ER is the closer the lesi
that angle is : and when that angle is of the same magnitude^
the contact of EM and ERis Jhe closer the greater the circle of
curvature is : for since TR is to TM as TK is to TQ, RM
(which subtends the angle of contact MER> is to TR as KQ
is to TK ; and, consequently^ RM is to KQ as the square of
ET is to the rectangle KTQ; so tliat, when ET is given, RM
Ls as KQ directly, and the rectangle KTQ inversely ; and when
KQ is given, RM is less in proportion as the rectangle KTQ
is greater* When BK touches the circle BQ at B, they may
touch it on the same, or on different sides of their common tan-
gent ; and the angle of contact KBQ may admit of the same
vliriety with the angle of contact MERin the former case. But
as there is seldom occasion for considering those higher degrees
of more intimate contact of the curve EMH and circle of cur-
vature ERB, we shall call their contact or osculation of the fame
kind, when, the'chord EB and angle BET being given, the
angle contained by the tangents of BK and BQ is of the same
magnitude.

S69. Cor. III. The curvature \b uniform in the circle only.
When the curvature of EMH increases from'E towards H, and
consequently corresponds to that of a circle gradually less and
less, the arch EM falk within ER the arch of the circle of
curvature^ and BK is within BQ. When the curvature of EM
decreases from £ towards H, and consequently corresponds to
that of a circle that is gradually greater and greater, the arcfai
EM falls without ER the arch of the circle of curvature ; and
BK is without BQ« According as the curvature of EM varies
more or less, it is more or less unlike to the^unifpml curvature
of a circle ; the arch of the curve EMH separates more or less
firom the arch of the circle of curvature ERB, and the angle

U 3 con-

N.

n

^^Q Of tht Curmture 6f Unts, Book I.

jbo^talB6<i ht the tfin^ents of BKF and BQfi at B ii greater or
less Thus the 5110% of curvature (as it ib called by Sir
Isaac Newtm in a treaUae lately published by the ingenious
Mr. Cokon) depends on the an^e conUined by the tangenU
of B£ and BQ at B ; and the measure of the inequability or
variation of curvature h as the tangent of this angle, the radius
being iiiven, and the angle BET being right ; for th^ index of
this variationis as the fluxion of the ray of curvature directly,
^nd the fluxion of the curve inversely : and we *all show after-
ivards (art. 386) that the fluxion of I'K when M comes to E
IS always to the fluxion of the ray of curvaUire in the invanabla
ratio of two to three. The measure of the angle of contact
ME?t contained by the curve a^d circle of curvatuare, depends
pot on the angle in which BK intersecU BQ, only ; but is as
ihe tangent of this angle directly and the square of the ray ot
curvature inversely (as wiil appear afterwards), when the tan.
irents of BK and BQ intersect each other in any assignable
luo^ie; and this measure observes other proportions when BK

touches BQ at B. ^ . .,

370. Cor. IV (Jig, 15ft). The rays of curvature of similar
orches in similar figures are in the sameratipasany homologous
lines of these %ures, and the curve BK cutsBQin the sameangk,

or the variation of curvature is the same. For let EM and em he
any similar arches situated as we described in art. 122, so that,
S being a given point, }f SM meet EM and em in M and m,
SM may be always to Sm in the same ratio as SE is to Se. Let
SM and Sm mcfet the tangents ET, et (which are parallel to
feach other, because the figures are similar and similarly si tnated)
In 2 and « ; and MZ shall be to tnt, TM to pn, and ET to et,
^s S£ k to Se. Therefore, since the rectangle MTK i» to mtk
Jls the square of ET is to the square of e#, TK is to tk, MK to
infc, and SK to Sk, as SE is to Sir ; and the figures SKB, Shb
are'Mke>;^ise similar, by ail;. 122. Therefore EB is to eb as SE
is to Se; and (because the angles BET, bet are equal) the rays
of curvature Are in the same ratio. The tangenU of BK and ft*
at B and ft are parallel, and cut the circles of curvatore BQ and
hqin equal angles : therefore the variation of curvature is the
samein similar archer. When the tangent of BK cut*the tangent

t)f

Plape3gLA3».K»/.I .

1*

I i

H

»■ \

«

r

> «

%

tact M£R andflier oootwed b^ ^im)^ W^^^ m4 ^<^ <^1^
of cturvntur^j «w reciprocelly ^^ f^Q a^i^orf Qf ^ ^J# of fnyfr?
Yf^twe^ or of wy hQilK>log<^up iilic» pf ^« %!iie9. W!>CT ¥JPi
tQ«Qh#» BQ^ iHit 90 9» npj^ to b^v^ ik^ ^^1^ qi^alfif e vi^ i|«
ib^^e iHQAsures aj?e reciprocally a^ tl^ie i^^ ^4i^ r^s 9f c^**
▼fttm^. When BK jto»ab^ ©Q, m# ^Nb^ ^he a^f WHTft^f
with it^ bM their con^t 19 of tb^ sef^ood cl,cgffe spjjr; ifiq»
Ibque meanires «re irecipipcfjly fia .tb# ^Hrtti f^oflr^ ^ t^ <v|y9

S71. Cor.V. Lettbect*yye]SMJi(/ga5S)i,fore>^^I9plp^^
• p«mboK£0(i4ijameterj £Tt}i^f#ngei4ftt|;{ ^bscjmeJtjbf
i^ot^i^le Goiib9Jped ^y TM and .th# pftriiifiot^ of if^e ^¥^^19
SB is equal to the sqiw)^ of £T> m *bf^ l^ffmKAf ¥1^> iij^f^^ir
Jow^j thftt TK 10 (Jway9 i^qmi %o ^ p^^LOMifix^ l^at i^ t]b^
cwie BK 18 a jrighi li^e i^aUd to (fe^ twg^nt £7, fmd 4j^t i^
ini^ilBectsEBioB^sothatEJI^Uequaltotbatpar^Bafter. 'QMror
jbne^ jf upon the diam^t^r of a jp^fkl^pla « ia,gk|. lin^^ip he Mc«ip
fcom E the v«rteac of this 4iw«}#r eflspl jo i^ ps^^fn^t^^ »
oif cle ERB 4escrihed pipoQ jt)n^ ^f^% )me a^ ij^ f^hor^^ that
touches <j»e poraboto at E> ^l^t l^ftbe cird^pf iHuryatjjW^. ^
oa^M the right line SlK wjt» ^e ^iifciie ]pQp ia 1^, qnl^ Tf^9
£ is the vertex of the figHre^ ^ jp^raholft ^i$ ^ ctfcjk ^
curvature (that case e^c^>ii^}» ^o^ p^pes witfaji^i the circle of
curvature wheait isproduq^.ta'^arfjs the vaiites:^ but iprifhou^i/t
vrhen produced the contr^iryx^y. ](^Jpuepfkpfusf|b^£iVIJ3f^^
4Uiycurve Enth hav« the s^necurv^^turejat £^ {ip^fupab^Uoaiibe
drawn^hroughEbetwiictEHH md £f9|ibhavu)git^^9paial^
rto EB; for^ sappp^i^ ihe rectangle mTk to be al^yi» eqf^ ^
the squf^e of £T» the cui^«//p ebaU pass ibrfH^gh B (by tt]|^
supposition) : and it i^ sbourn^ 4^ the very ^^aqifi ^9^r^ t}^
thj^ proposition was de^ions^ai^d^rthAt ai^ olhear p«jr^}(^^
sofibed through Eisbfdl-^^r £41 M^ithoiit botb^EM! m^ ^m, v
witbitt tbeai both. JH^ence tb^ ohprd pf curvs^tfire ffi 4^ ai^
oprw EffiA i$ ftnwdj wh^ j^ pwsJ>Ql^ is djateifiiwi^e^ t}i^
.touches it so closely ihfftjko parabpia qsok paf» betfirfi^n ib^*
S7«. C<w.yi. Let BM0fea44)be#nbraefbote, APWMgl AJi tl^e

; J^tbe.<»j«eW:^Twfi«^t t^a»;w^

U4 let

Sl« Of the Curoature of lAnes, ' Bookl

let the angle EVB be made equal to the angle HAP contained
by the asymptotes^ and VB meet EB parallel to the asymptote
within the curve in B : then a circle described upon the chord
EB that touches ET shall be the circle of curvature. For^ in
this case^ the point K is always found in the right line VB ;
because^ if MT produced meet EP parallel to AH in Z, and
T6 parallel to VB meet EB in 6, then^ because of the similar
triangles EZT^ £T6^ the rectangle contained by EG and ZT
fthall be equal to the square of ET^ or the rectangle MTK ; and
TK shall be to EG as ZT is to TM, or (by the properties of the
hyperbola) as PZ is to EZ, or VT to TE, or GB to EG ; there-
fore TK is equal to GB^ and K is always found in the right
line VB. Because VB cuts the circle BQ^ unless when the
togles EBV and BEV are «qual^ and consequently £ is the
vertex of the figure^ it follows^ that the hyperbolacutsthe circle
of curvature in all other cases. When ET the tangent of any
xmrve EM and BV the tangent of the curve BK^ are given^ an
hyperbola may be described through E^ that shall have the
same curvature with EM^ and the same variation of curvature^
or a contact with the circle of curvature of the same kind^ by
producing those tangents till they meet in V^ and describing
through E an hyperbola that has an asymptote through V
parallel to EB» and another asymptote through u (Eti being
taken equal to £V upon VE produced beyond E) that consti-
tutes with the former an angle uAV equal to EVB^ or that
makes the angle EkA equal to EBV.

373. Cor.Yll. IjstEMH(/ig. 155)beanjrconicsection,ETthe
tangentat E^ HIatangent parallel to EQthatmeets ETinl.and
kt EMH meet EB in G. Take EB to EG in the same ratio
as the square of EI is to: the square of HI^ or (when the conic
section has a centre) as the square of the semidiameter Oa pa-
rallel to ET is to the square of the semidiameter OAparsJlel
to EB^ and a circle described upon the chonl EB that touches
ET shall be the circle of curvature. For let TM meet the co-
nic section again in m ; and^ since the rectangle MTm is to the
square of ET, or the rectangle MTK, as the square of HI is
^lo the square of EI, Tm is always to TK in the same ratio ;
and^ oensequently, the point K is always found in a coiiio ^o-

- > tion.

Chap. XI* <mJ its Variation: 315

taon^ that meets EG in B^ so that £B is to EG as the square of
EI is to the «]uare of HI. Let GV the tangent of the iBrst co-
nic section EMG at G meet £T in V ; and, because Tm is to
TK in an invariable ratio, and their fluxions are in the same ra*
tio, it follows, from prop. 14, that BY shall be the tangent of
the conic section BKF at B. Hence it easily appears whether
the arch EM of the conic section falls without or within the
circle of Curvature, and the different degrees of osculation be^
twixt the conic section and the circle of the same curvature
may be compared together. For let any conic section EMH
meet, the chord of curvature EB in G, draw the tangent GV
meeting ET in V, join BY, and, the angle BET being given,
the nearer the angle EBY is to BET, or\ the nearer the ratio of
EY to BY is to a ratio of equality, the closer is the contact of
the conic, section EM and of the circle of curvature at E. Let
the angle EBv be made equal to BET, and let Bv meet ET ia
V, join Gv, and of all the conic sections which can be described
through E and G having the same curvature at E with the circle
ERB, that which touches th^ right line Gv has the closest con<-
tact with the circle : when BET is a right angle, or EB is th^
diameter of the circle of curvature, EG is in Uiis case the axis *
of the conic section (because the angle EBr is a right one),
EB is the parameter of this axis, and when the points G and B
are on the same side of £, EMG is an ellipse, and EG is the
greater or the lesser axis, according as EG is greater or less
than EB.

374. Cor. YIII. The propositions relating to the curvature of
the conic sections, which are delivered by authors' on this sub- '
ject, follow from what has been demonstrated. 1. ^^en the
chord of curvature E'B(fig ld6)passedthroughOthecentreof the
conic section, A coincides with E, EG is a diameter, OA and
Oa are conjugatesemidiameters; and, since EBis to EG as the
square of Oa is to the square of OA (or OE), or as the para-
meter of the diameter EG is to EG, it follows, that EB the
chord of the circle of curvature is equal to the parameter of
EG the diameter that passes through the point of contact. This
was shown of the parabola, in art. 37 1 • 2. Let C be the cen-
tre of the circle of curvature, and Cb be perpendicular on the

dia'

1

914 OftU Cwrvahtn of ldn€9, BotkT.

dtniMlcr EG mi ; kt 0« meet EC in N : vxi, the tiianglev
EOIi, £Ci being similar, the rectangle CEN is equal to the
scctattgle OBJb, which is equal to the square of O^, because
Ei i»OMe half of the parameter pf EG. Therefore the square
of the semidiameter Oa is U> the rectangle contained bj Oo
iaid EN> or the invariable rectangle contained by half the
%t9mwti€se and half the conjugate axis, as the ray of curvature
C£ is to Oa ; and the cube of the aemidiameter Qa^ that 19
eot^gate to OE which passes through the point of contact, is
o<(«ftltathe solid containedby theray ofcurvatureand thatinva-
liable rectangle, asiadiown by Mr. de Moivre, Misc. analyt. p.
ms* 3. htt EH (Jig. 157) be an ordinate to an axis of thie co*
Bic section, and^ £i being in this case equal to HI^ £B shall be
eiqsal to £6^ or B coincide ivith G: from whidi it followa, that
ITCH be drawn from E perpendicular to either axia^ and the
angle HEG be made equal to HET on Ae opposite ^de of
B£^ then E6 riiatt meet the conic sectson in the point G where
tte i»rdk of cut vatare and come aection intersect each otiter»
Sr^ Cof . IX.O^g. 158^ n. 1 and£.) Tothese wemay add thefol-
lowing pn>pertie8of theciorcleofcnrvatasehdongingtoauy point
ofaeonicBoctioiii. 4. InthedlipseorhyperboU,letCXf,theaemi*
AattieterpafaHelto ETthetaagentof the section atE,nieet£B»
any chord oftbe circle of curvature^ in R; bisectEBinA; and
the rectangle REft skaU be equal to the square of O0. for, let
tile ^ameter through the point of contact aoieet the ciicle of
curvature again in e, join Be ; and^ the angle &B being equal
to BET or ERO, the triangles E&, EOR are simUar, EB is to
£r as EO i» to ER, and the rectangle REB is equal to the
tec tangle OE^. But Ee is equal to the parameter of the diame-
ter trough E, by the first property of the circle of curvature
in the last article : therefore the rectangle OEc (or REB) la
«qnal to twice tliesquareof Oa^ and the rectangle REft is equal
to the square of Oa. When £B passes through the fiocut, £R
is equal %o half the transverse axis ; therefore the chord of the
circle of cwrvature that passes through the focus> the diameter
<*Dnjttgale to that which passes through the potat of oontoct^
and thetransverse axis of thefigure^ arein continued proportiott.
It appears, likewise, that when the section is an ellipse, if tbe

circle

Chap. XL and iH Venadm. f f S

eirde of cnrvatme oi E meet Oa in df the square of Bd Aall be
equal to twice the square of Oa ; therefore Ed is to Oa i& tibe
inyariable ratio of the diagonal of a square to its side^ or of the
square root of two to unit : and hesce when the point of con^
tact E and the semidiameter Oa are given in position and mag^
nitude, the centre of curvature is readily determined. 5. Tbd
right line EG being supposed to meet the conic section in any
two points E and G^ and the tangents at those points to intw-^
0ect each other in V^ let £B be bisected in 6^ join Yh ; then the
angle EVb shall be equal to the an^e GEO^ or to its supple^
jnent to two right angles ; and a circle through E, V^ and b
shall ahvajs touch the dianieter that passes throu^ £. FoTj ki
the eUipse and h jperbola^ if Op parallel to EG meet the tan^
gent EV in p, the rectangle contained by Y^ and Ep (or R0|
shall be equal to the square of Oet^ or to the rectangle RE&;
and EV is to E& as RE is to RO. Therefore^ since OR is pa^
rallel to EV^ the triangles REO^ iEV are similar^ the ai^U
EV6 is equd to REO, and a circle through E^ V, and 6 touchei
EO. In the parabola, let EG (^. 158,n.3)bebisected ing; and^
since Eft is to Eg as the square of EI is to the square of HI>
(by art. 373), or as the square of EV is to the square of 'Eg ; H
follows, that the rectangle &% is equal to the square of £V^
and the angle £V6 equal to EgW or GEO* Hence when, in
any conic section, tlie tangents EV, GV «re given, and the
diameter through E iv given in positron, the point b and the
centre of curvature are readily determined. W^hen any two
points in a conic section,, as E and G (fig. 158, n. 1, 2, and S),
and the tangents at these points EV and GV, with the circle
of crurvature belonging to one of them, as E, are given, the
section is determined by bisix;ting £B and EG in b and ^, join«
ing Vb and Vg, and making the angle 6EL equal to 6V£,so that
EV and EL may be on opposite sides of £6 : for EL dmll be a
diameter of the section, and if it intersect Vg in O, then O
shall be the centre of the figure ; but if EL be parallel to Vg,
the figure is u parabola. When b and g are on different sides
of E, the figure is an hyperbola ; and when t^iese points are
on the same side of E, it is an hyperbola or ellipse, according
as the angle SNb is greater or less than E^V. When two points
£ and G of a conic section are given, with EV the tangent

at

1

S 10 Of the Curvature of lines, BoA L

at E, and the points B and D where the circles of corva*
ture at E and G meet EG, the tangent GV is determined, by
drawing fix)inGtothe tangent EV aright lineGV,so thatGV
may be to EV in the subdupUcate ratio of GD to £B. For if
HI (fig. 165) the tangent at H meet GV the tangent atO in i.
Hi and HI shall be equal ; and if the circle of curvature at 6
meet EG in D, GD shall be to £B as the square of Gi is to
the square of EI (by art. 373), or as the square of GV is to the
f^uare of EV (fig. 159, n. 1 and 2). 6. Let EB be the chord
of the circle of curvature that passes through S the focus of the
conic section ; let BX paraUel to the tangent ET meet EX
perpendicular to ET in X ; and if XZ be perpendi<?ular to EB
in Z, then shall £Z be equal to the parameter of the transverse
axis of the figure. For let EK be equal to the half of this
parameter; then since EB is to EZ as the square of EB to
the square of EX, or as the square of ER (which in this case
is equal to half the transverse axis of the figure) to the square
of EN, and, consequently, as the square of Oa to the square of
halfthe conjugate axis, or (by the fourth property of the circle of
curvature) as the rectangle 6ER is to the rectangle KER, that
is, as &E is to KE ; and, since EB is equal to S6E, it follows,
that EZ is equal to 2EK {fig. 158,n. 3). In theparabola, letSP
perpendicular on the tangent at E meet it in P, and A be the
vertex of the figure ; then EB shall be to EZ as the square of
SE to the square of SP, or as SE is to SA ; and sinoe EB is
equal to 4SE (art. 37 1 ), EZ is equal to 4SA {fig. 1 59, n. I and 2).
Hence, when the focus S, point of contact E, the tangent ET,
and EB the chord of the circle of curvature through S are
given, the principal parameter of the figure is readily determine
ed ; or, when the rest are given, the centreof curvature is easily
found. 7- liCt S and/be the two ybci when the section is an
ellipse or hyperbola ; let ON parallel to the tangent ET from
the centre O meet EN perpendicular to ET in N ; let C be the
centre of the circle of curvature at E ; join SC and/N : then
shall the angle ESC be equal to EN/. For the rectangle CEN
is equal to the square of Oa (by the fourth property), the square
ofOa is equal to the rectangle contained by SE and/£;
therefore CE is to SE as/E is to EN : and, since the angle
SEC is equal to /EN, the triangles SEC, ^Ef arc similar,

and

Cbap. XI. and lis Fariation. S17

and the angle ESC is equal to N]^. Hence the centre of

' curvature at £ is readily found when the two foci are given.

8. If £Q be taken upon £B, that chord of the circle of curva«

^ tare which passes through tlie focus S, equal to one fourth

part of £B; then SQ^ S£^ and the transverse axis of the section^

shall be in continued proportion. For^ since the rectangle REfr

is equal US the rectangle contained by SE and /E^SE b to EQ

(or one half of £6) as 2ER is to /£ ; irom which it follows,

that SQ is to SE as SE is to 2ER> which is equal to the'trans-

verse axis of the 6gure. Hence, when the focus S, the point

of contact E, and tangent ET with EB that chord of the circle

of curvature which passes through the focus are given, let EQ

be one fourth part of EB, draw SP perpendicular to ET in P,

let Pr bisect SE in r, let PO be taken upon Pr (the sameor

the contrary way from P with Pr, according as EQ is less or

greater than £S) equal to one half of the third proportional to

SQ and SE; and O shall be the centre of the figure. Accord*

ing as EQ is greater or less than ES, the figure is an hyperbola

or ellipse ; and when EQ (Jg. 158, n. 3) is' equal to ES, it is a

parabola, the vertex of which is determined by makingthe angle

PSA equal to PSE, and PA perpendicular to SA in A.

376. Cor. X {Jig. 160). The variation of curvature at any

point of a conic section, is always as the tangent of the- angle

contained by the diameter that passes through the point of

contact, and the perpendicular to the curve at the same point.

Thus the variation of curvature at £ is as the tangent of

the angle GEO, EG being a perpendicular to the curve, and

£0 a diameter. For if the tangent at G intersect the tan-
gent ET in V, and b be the centre .of curvature, the angle

EVb shall be equal to GEO, or to its supplement to two right
angles, by the fifth property of the circle of curvature demon-
strated in the last article. The variation of curvature at £ is
as the tangent of the angle EVB, or of EVb (because tlie tan-
gent of the latter angle is always one half of the tangent of
the former) ; and, consequently, as the tangent of the angle
GEO. Hence the variation of curvature vanishes at the ex-
tremities of either axis, and is greatest when the acute
angle contained by the diameter OE and the tangent TST(Jig.
l6i) is least. When the section is a parabola, and S is the

focu».

1

518 Of the Cwnrntwif ef Lines, Book L

ftcu^|ji# aagle GJEO is equal tp QES; and did variation is a»
t|)e tangent of the Mgte contained by tbe rigjat Une drawn from
tt^e point of contact to tbe focxuj and the perpendicular to the
eyirve. Hence> wben tbe point of contact E^ the tangent EV,
w^(h ^b^ curv^i^ at £ and its van«^ion are givea^ tbe parabola
IS detennined : £6r^ BE and £V beinggiven^ bisect EB in i> joia
Vi^^m«J(e the angle B£b equal to&V£,andiet Ebn^eet the circle
of 4Hiryati)re in b ; take £S upon £b equal to one fourth part
of JBb : and S fibaU be the &cus of the parabola. If £B meet
tjba axis of liie parabola in N, and £L be perpendicular to tbe
axis in h, the variation of curvature shall be as the tangent of
tii^ wgle ENL^ or as the ordinate £L directly^ and the para-
]|ieter of t|^ spp^ inver^ly . The joieasure of the »)g!e of con-
tact contained^ t^3^ the paiab^Ia and circle of curvature at any
point Ey is as £L the ^dinaJ^e to the axis directly, aad the cube
of ^£ the distauice of £ fropi the focus iavereely .
.377. Cor. XI i^. l€3). When EBdoesnotvueet with theeurve
FK^ but is ita asymptote^ any circle being described touching
!|£T in £j a greator circle ^hali always pass between it and the
«»i;ve EM; and the greater thi&circle is^ the closer shall its oon«
tact be with the cufve EM. For^ siiice tibe curve FK pdroduced
paflses without any circle EQP how great soever that can be de-
scribed throu^ £4 EM mv^t always pass betwixt £R and the
taiigeut ET. This is t^e case in which the curvature is sfud to
);>e infinity little (being less than that of any ciirc^)^ or tlie
my o( curvature infinitely great. Tbe point E in the curve
HMEi^ af>oi2it of contrary flexure when the two hyperbolic
branobes qf the G^rve FK proceed along the asymptote EB on
difleuestsides of it and with oppoaite.difecftions, as in the com*
mon hypevbolft. But when these hranohes ane on the aameside
of (heasymptote^and proceed along it with opposite dieeotiona,
E is a cuspid of the first hind ; when they proceed with tbe
same direotion on di&oent sides of the asymptote, the cunw
EM has its concavity turned tiie same way on both sides of £;
when those branches proceed, along the asymptote on the sama
side of it, and.with tbe same directioii^E is a cuspid of the se*
oond kind t and in all those caacsithe curvature at Eis less than
that of ady oirole. Of tbe fij^Looae we have an exam{^ in tha

vertex

I

Plate»ir/%jrJW Vol J

. I

Chap. XI. otid ks F^riaiim* ^19

i^ertex of lihe osAic f lurabola, wherein like cAie ^ ET beiiq;
equal to the solid contained by TM and a given square, the
MctABgle contained bj TK and ST mnsi be equal to this pyeu
iqmffe, and the curve FK is a eomHioa hyperbola that faai
EJB and BT for its asymptotefl. The curvatare is of the weame
kind at the veitex c^any paralxJa wherein TM is as any powa
of £T whose exponent exxseeds ^, for FK in aH those cases h
an h3rp«rbola of which EB is an asymptote.

573. Cor. XII. When the curve FKO^. \€S^ passes <hfoa^
£^ no ci^rde ean be described through E sosmal) but a lesseircte
aha!) pasK between it aind Incurve EM; and the less ihisieirde
M^ the doser diall its contact with EM be. For^ smce the corre
fSVL paases within any chicle that ean be described thnmgh C
on the mme side of ET^ the arch EM of the cunre EMtf ia
always within BR the arch of any .ehc^Ie ERB. In this casi^
t^^cmisethe ciirvaitui«sorpasses thotof any circle^ rt is said tobe
wiiaitely greats or the ray of cnrvaitare to be infinitely little.
Whtn FK passes tbiough E, tt always touches EB there^ and
if it has a continued curvature at E,the curveHME has acns^
4»f f be first kind at £ ; if FK has a point of contrary ilexme afc
E of any kind^ HME has likeime a ^mnt of contrary flexare
^£; ifFKhasacuspidaf4be«r5tkindatE, HMEhasits
^eoncarity turned the same way^on both sidesof E ; and if FK
iiavea caspid of the second kindat B^ HMEsbali have a cuspid
«f the same kind there : and in sill those cases tbecnrvaturedf
ilME at £ surpasses that of any circle, of Whatever kind the
•ieuTVatuxe of FKE at £ be. Of the first, we have an example
^attfae vertex or cuspid of thesemicubical parabola, in whidrtbe
«cttbe of ET being equal to the solid contained by the square of
'TM and an inyariable right line, the square of TK is equs3 to
ifae lectangle contained by ET and the same invariable right
'line; »id therefore FK is a common paMbola that touches 1^
■in £. When FKB intersects EB in two points, neither of which
4)dinctde with £, then £ is nota cuspid, but a point where 't^ro
■atfohes of the curve HME touch each other that are both :con-
HHHiod Oft each side of E. Of thi& kind there are no points inthe
/Unesof any oider beneitth the fourth, as there areno poinks.of

1

520 Of ihe , Curvature i^f Lines, ' Bod(L>

contraiy flexure^ or cmpidsj in the lineft of ftfiy order beneath
the third.

S79* Cor.XIIL SupposenowtbatEMH0%*l63)ift^7lu^
of the third order; let ET the tangent at E meet the curve in

A, EG any right line through Emeetit in the points ^ G> andg,
and TM parallel to EG meet it in M> m, and m : then^ if EBbe
taken upon EG on the concave side of the arch EM, so that the
rectangle AEB be to the rectangle G% aa the solid contained
^ AT and the square of ET is to the solid contained by TM«
Tfij and Tm, a circle described upon the chord EB so as to
touch ET shall be the circle of curvature at E. For these
solidsarein an invariable ratio to each other when ETand EG
are given in position, and TM is always parallel to EG, by the
properties of the lines of the third order ; and^ since the reci-
angleMTK is supposed equal to the square of ET> it follows,
that the rectangle ATK is to the rectangle mTm in the same
invariable ratio ; therefore the curve FKB shall meet EG in

B, so that the rectangle AEB shall be to G£g in the same ratio.
In all the lines of this order the curvature at the confine of con*-
trary fiexture is less than the curvature of any circle, or isof that
kind which was described in cor. 11. Por when E(/Ig. l64, n. 1)
is at this confine, the tangent ET meets the curve in the point
£, only ; the cube of ET is to the solid contained by TM,
Tm, and Tm in an invariable ratio : therefore the rectangle
ETK is to mTm in the same ratio ; i^onsequently EG is an
asymptote of the curve FK in this case, and (by cor. 1 1) th^
curvature at E(fig. l64, n. 2) is less than that of any circle. It
is the contrary when £ is a cuspid in any lineof tbisorder : forthe
curvature at the cuspid is always greater than that of any circle,
or isof the kind described in the last corollary: because, in this
case, while T approaches to the cuspid £, the ratio of £T to Tm
may become greater than any givenratio ; consequently, theratio
of Tm to TK may likewise become greater than any given ratio,
and whenm comes to g, K must come to E : therefore the curve
FK passes through E,and the curvature at the cuspid Eis greater
than that of any circle, by the last corollary. In these lines
(fig. 163) the variation of curvature is found by determining
Ae tangent of the curve FKB at B when GET is a right

angle ;

I

Chap. XI. end its Variation. 321

angle ; and it is easy to show, that if GR and gr touch the
curve at G and of, and meet EA in R and r respectively ; an!i
if BV the tangent of PKB meet EA in V ; then EV shall be
to EA as the rectangle REr is to the sum, or difference, o/ the
Rectangle Ri\r and the square of EA.

• seo. Cor. XIV. When EMH (Jig. 149 and 150) is a geiv
metrical curve, BKF is likewise geometrical'j^, since TM, TE,'
and TKture in continued proportion ; und since the pdJutB is
determined by the intersection of a geometrical curve wHl) the
right line EB, it follows, that, when the dngle BET is given^
the curve in which B is always found is likewise geometrical ;
and, according as EM H iis geometrical, or mechanical, i!be
curve in which the centre of curvature of EMU'is always found
is geometrical, or mechanical. By this proposition and the pie-
ceding corollaries, the curvature of EMH and its variation^
with the degree of contact of the oarve «ind circle of curvature,
may be determined when EMH is geometrical. We now pro-
ceed to llie theorems by which these are deWmlned with
equal facility when it is mechanical.

381. Let Em (fig. 1 65) be a common parabbli, ET the Jan-
gent at 'E, Erf the diameter through E^, EB th^ barameter of
this diameter, and let any right line Tm partner t^o td nieet
the tangent in T and the parabola in m. Let e/a, right line given
in position meet £J in £ and Hm in/; thea,if^ be taken i^n
fm so that the toiangle efn be always equal to the rectaja^e
contained by Tm and a given ri^t line PG, the point la^ shall
be always found in a right line en given in position. Fof,
since the rectangle contained by Tm and £B is always equal
to the square of ET (by the known property of the pariJwla),
the triangle efn is to the square of ET as DG is to EB. Becau^
£T and e/are given in position, aud/T is airways parallel 1p
eE, the square of ET is to the square of ^always in a given
ratio ; consequently, the triangle efn is to the square of ef, and
fnistoef/ina given ratio : therefore en is a right line given in
position. And, Conversely, when ET, ef, and m are given in
position, if the rectangle contained hyTm and a given right line
DG be always equal to the triangle efn, the point m shall be in a
parabola that touches ET in E so as to have its diameters

VOL. L X parallel

322 Of the t^uroaiure of Lines, Book I.

parallel to Tm, and the parameter of the diameter through £ in
the same ratio to DG as the square of £T is to the triangle
€/}i; for supposing EB to be to DG. in this ratio^ the rectangle
contained by Tm and EB shall be equal to the square of ET«

382. Let the base AP floir uniformly^ and its fluxion be re-
presented by any given right line DG ; let PN the ordinate of
the figure DeNP always measure the fluxion of PM tlie ordi-
nate of DEMP^ as in prop. 20^ let en the tangent of eN at £
and ^parallel to DP meet PN in n andy; and £T the tangent
of EM meet it in T : then^ when DP becomes equal to DG,
fn shall measure the fluxion of the ordinate De (by prop. 14),
or the second fluxion of DE 9 and ET shall measure the
.fluxion'of the curve HE.

PROP. XXXIII.

The hose AD (Jig. I65) being supposed to flow unifotTnly, let
the second fluxion of the ordinate DE be to the flujcion of
the curve WE as the fluxion of the curve istoEb; and ifER
he taken upon DE from E on the concave side of the curve
equal to iEA, the circle of curvature at E shall pass through
B.

Let the rectangle contained by Tm and the given right line
DG be always equal to the triangle efn, and, by the last article,
the pomt m shall be always found in a parabola Em fhat
toudies ET in E, whose diameter through E is Ei2 ; and the
parameter of this diameter must be equal to EB, because it ]<^
to DG as the square of ET(or the rectangle contained byy>i and
Eft when DP becomes equal to DG) is to the triangle efn (or
the rectangle contained hy fn and one half of e/; or DG);
that is, as EB is to DG. But no parabola can be described
through E betwixt the curve EM and the parabola Em. For,
let Ett be any other parabola described through E, touching ET,
and having Ed for its diameter; let TM meet this parabola in
u, and. the rectangle contained by Tti and DG be always equal
to the triangle ^x, then the point x shall be in a right line ex

given

Chap. 5cl. and its Variation. 323

given in position^ by the last article ; if £b b^ the pi&rameter
of the diameter of Ett that passes through "E, the triangle ^
•hall be to the square of ET as D6 is to £b, and (because
the square of ET is to the triangle efn as EB is to DG) the
triangle {/x to e/h as EB is to Eb, so that fs is toyTt in the
same ratio. The rectangle contained by TM and DG is always
equal to the trilineal area f/*N, by what was shown in prop. 20.
Therefore the right lines TM, Tm, and Tt* are in the same pro-
portion to each other as the area ef^, the triangle e/n, and the
triangle efx. But, because tn is the tangent of the curve eN>
if we suppose PM to move towards DE, the area cTN, and
triangle </n'shalleither becomeboth greater or bothless than the
triangle tfx (by art. 181), according as fn is greater or less
than/r, or Eb is greater or less than EB ; consequently, TM
and Tm shall either become both greater or both less than T«
whileM approaches to E ; and therefore the parabola Etf cannot
pass between the curve EM «nd parabola Em, In the same
manner, noother parabola can pass between EM and Em; but
an indefinite number of parabolas can be described between EM
and any other parabola Eti, as an indefinite number of right
lines may be drawn through e betwixt en and ex within the
angle nex. Therefore the curve EM and the parabola Em have
the same circle of curvature, by the latter part of art. 373 ; and
this circle is that which is described upon EB so as to touch
the right line ET, by the same article.

383. Cor, I. As all curves that pass through E have the
same tangent when the first fluxions of thdr otdinates are
equal, the fluxion df-the base being given ; so Aey have the
same circle of curvature at E when the second fluxions of their
ordinates arejikewise equal. Thus, the motion with which*
the base flows being given, the position of the tangent depends
on the motion with which the ordinate flows, and the flexure
of the circlet>f curvaturc depends on the acceleration or retar-
dation of this motion. When the angle BET and the fluxion
of the base are given, the ray of curvature EC is reciprocally
as the second fluxion of the ordinate ; because EC is to £6
(which is reciprocally hsi/n) as ET is to DG. When the fluxion
of the base is consttinl, tke solid cootained by EC the ray

X2 of

i

32* Of the- Curpafurc of Lines, - Book I.

df cuiratare, DG whi0h measiiiTes the fluxion of the base, ani
/h tiftit mensur^ the second fittxiofi of the ordinate^ is alvaj-s
eqtlftl to the cube of £T w^iv'h meiLsures the flaxioii of the
curve. Ifthe^curveER dfig. l67)begeneratedfromerin tliesaine
Irianner as EM is generated itom eN (that is^ if the rectangle
tontained by DG and TR be always equal to thearea t/r), the
contact of EM and £R shall.be always one degree closer than
that of €N|aad er. Thus if eN and er pass through the same point
0, EM and 'ER shall have the same tangent; when they have
tlie same tangent^ EM and ER have the same carvature ; when
is and er *have the ^anle curvature^ the contact of EM and
ER with their common circle of curvatase is of >tbe same kind,
dr tlhey haite the same variatton of curvature^ and so on.

384. CcT. 11. The sacood fluxion of the curve HE (Jig. i66)
k to the first fluxion of the curve jrs the fluxion of the ordinate
P£ is to E6: and^ S being any giv^npoint in the line Hd, i£
SK be always peipendicukur ^ thie tangent of H£ in K, the
iuxion of the perpendicular SK ^haUbe to the flaxion of SE as
SK as to ISJb, orms SE is<to tbe ray of curvature EC. For^ let any
Ordinate pm meet the cusve EH in n^ and eN in n; let nf pa*
mflel to the base meet 6M an'f> {^d S/e be peipepdicular to the
ttingentat miniik: then I^'axid I>f i^uJl bq ]^aUel to thetan-
^ents of the cocve HEm a;t'E^d vft respectively^ and shall
measure the fluxions.of thfe al^ehes HE and Hm^ the fluxion of
the base being measured by J>Gj pr ef Let ny be perpendicular
«D lyin y ; and since tjbe fluxion of D^ is n^easure^ bf/n, the
fluxion of iy^(or l^e second j^uxionof the curve HE) shall be
measured by j^i>4>y art. 199, But j^js to Jh as/i is to B/^ or
(bythifipnip<HiliQB)a8ltriS'toEii». Tliesefbre the second fluxion
of tUeronrfc (which ia measured \$yfy) is to the fluxion of the
oihrve <6r D/) as the flu^n of the ordinate (or Dc) u to ££.
This likewise app^rs from art. 96» because the ^fluxion of the
square of £T> mBfi is eq^al to the fluxion of tbe^qui^re of Tk,
DG being hivariabki« Let the angle SKx be made equal to SET
OB IVG/ andJC^r shall be the. tangent of jtbe c^rve Kk that
paisedal^ajr^ through t^^ioi^rsections of tb^twgentsof HEM
and :khe perpendicul^afrQU^ S^.^y ^^' ^1 !• I'be a^gle KSi is
flqnalto/DQ aiidrtUeaug«dal:vek>oijty of SK about S is equal i9
> the

Hate Xim[Az,SU\a.l

t

i

Chap. XI. ttnd its Variation. 3^

the angular velocity of ly about 1>. Therefore the iktxion afi
SK is to the fluxion of ly as SK is to ly,* and the fluodon oP
SK i^ to SK as the fluxion of D/is to Df, that is (by what has'
been shown)^ as the fluxion of DE is to E&« But the fluxioti'
of SE is equal to the jluxion of DE. Therefore the fliixion of^
SK is to the fluxion of SE as SK is to Ub, or as SE is to EG.
385. Cor. III. Let S be a given point iik the line J)d as in
the4ast article^ EV and EM any lines through E that meet'
pM^ parallel to SD in V and M ; lef Sp parallel to DP meet
PM mp ; join SV and SM ; and the difference of tlie second
fluxions of SM and pM shall be equal to the diflerence of the
second fluxions of SV and pV when M and V come to E.
For the diflerence of the squares of SV and pV is equal to the
diflerence of the squares of SM and pM : from which it fol-
lows (by art. g6 and 99), that, since the first fluxions of SV
and pV become equal, and the first fluxiotis of SM and jjM
become likewise equal, when V and M come to E and SV,
pV,SM,pM then coincide with each other, ihe diflTerence of
the second fluxions of SM and pM is to the difference of the
sl^cond fluxions of SV and pV at that term, as the sum of^
SV and pV is to the sum of SM and pM, and' therefore in a
ratio of equality. "VMien the base flows uniformly, and'EV is a
right line, the second fluxion of PV vanishes j therefore the
second fluxion of SV is in this case the same'whcn V dets out
from E in any right line that does not coi/icide'witfa ES, and'
is equal to the difference of the second fluxions of SM and'
|)M at the same term, whatever curve be deseribed by the point
M, Vl^hen EV is a circle that has its centre* in S, the second'
- fluxion of the ordinate PV is equal to the same difference of
the second fluxions of SM and pM, when M comes to E, be-
cause in this case SV has no fluxions of any prder whatsoever :
and if Eu any right line through E given in position meet
PV in ti, the second fluxion of Su shall be equed to the second
ffuxion of PV, the ordinate from the circle, when V and u
come to E. It appears, likewise, that the second fluxioo' of SM
is the same when M comes to E in aTl lines that have the same
curvature at E, and is measured by the diflerence beti^ixt /n
wbich represents the second fluxion of DE^ and a third propor-

XS tionaj

5a6 Ofth^ Curvature of Una, Book I.

tiopal \o SEl and D6 (which rapreBents the invariable fluxion
of AD) ; because the flaxion of the circular arch EV becomes
^Qfd tQ the flqxion of AD when V comes to E^ and the se^
QOQd fluxion of PV the ordinate of the circle is to DG as DG
18 to SE the radius of the circle^ by this proposition.
S86. Cor. IV.Therestremainingasin art. 3844etEB(/!g,l67)
be perpepdicular tp the tangent ET and to ADj aad siace^
which me^ures the fluxion of Ef (or the secopd fluxion of the
curve HE) vanishes^ the fluxion of the rectangle contained by
fn and Ep (which in this case coincides with E6) shall likewise
vanish^ because that rectangle is equal to the square of £T ;
and the fluxion of EC the ray of curvature is to the fluxion of
fn as EC is to/;i^ by prop. 3. I^et er be ^ parabola that has
the saxz^e curvature with ^N at e, and the rectangle contained
by TR and DG be always equal to the area efr^ as in art. 383,
Let the rectangles MTK, wTQ, RTA be each equal to the
square of ET, as in art. S66, and let TA; meet 6f the tangent
of B& (or of BK) in t. Then^ the fluxion of the base being
constant, the fluxion oifn shall be represented by 2»r, or 6otR,
by art. 255. But, since Tm is to TR as TA; is to TQ, 6mR,
(which measures the fluxion oifn) is to 6A:Q as Ttn is to Tk ;
and, therefore, as the rectangle mTQ is to the rectangle con«
tained by T^ and TQ, or EB. The rectangle nfTQ is equal
to the square of £T, or (by this proposition) to the rectangle
contained hyfn and E6 ; consequently, the fluxion ofyji is to
6AQ HAfn is to ST/p, and the fluxion of the ray of curvature
* EC is to 6*Q as EC is to 2T*, or as EB is to 4TA ; from which
it follows, that the fluxionof the ray of curvature is measured by
three halves of Qt, the Quxion of the curve being measured by
ET or BQ ; and that the variation of curvature, according to
Sir luiac Newton (who measures it by the ratio of the flux-
ion of the ray of curvature to the fluxion of the curve), is
measured by three halves of the tangent of the angle QB^ in
which tl^e curve BK intersects the circle of curvature, as was
observed in art. 369. But, since the curvature itself is recipro*
cally as th^ ray of the circle of curvature, if we shoi^d therefore
ipeasure its variation by the ratio of its fluxion to the fluxion of
the curve, this variation would be as the tangent of the angle QB<

directly.

Chap. XL and its Variation. Sil

directly, and the square of the ray of curvature inversely, and
would be as the measure of the angle of contact contained by
the curve and circle of curvature ; because, when a quantity ^
is always inversely as another quantity B, its fluxion is as the
fluxion of B directly, and the square of B inversely. How-
ever, to avoid confusion, we have conformed to Sur Isaac Ncoh
tbn*s explication of the variation of curvature.

387 . Cor.V. When (fig. 1 65) en the tengent of eN is parallel to the
base, and the rectangles contained by DG and TM and by DG
und Ttn, are respectively equal to the areas DPNe,DPiie, as in
prop. 20, then Em becomes a right line, and the curvature of
EM is of the kind that is less than the curvature of any circte^
which was described in art. d77« In this case, the second flux*
ion of the ordinate D£ vanishes ; and (provided the curve be '
continued on both sides of DE) if the number of the fluxions
of D£ of successive orders that vanish be an odd number^ then
E'is a point of contrary flexure ; but if it is an even number^
E is not a point of that kind, as was shown in art* S66,

388. Cor. VI (fig. 1 66). The same things being supposed as
in art. 384, it follows, from art. 202 and 208, that the velocity
of theangular motion of the tangent of HEm (or of SK about S)
whenm sets out from E, is the same when the point n describes
any curve that has the same tangent at e, or when m describes
any line that has the same tangent and curvature at E^ the flux-
ion of AD being given. Therefore the angular velocity ofihe
tangent of EM at E is equal to the angular velociify of the tan-*
gent of the circle of curvature, or of the ray of curvature about
tlie centre C, the fluxion of AD being given, because the angu-
lar velocity of the tangent of a circle is equal to the angular ve-
locity of the ray drawn from the centre to the p<rint of contact^
by art* 18. In like manner, if B (Jig. l68) be any point in the
circle of curvature, and the right lines EM, BM revolve about
E and B so that their intersection M describe the curve HME^
the angular velocities of EM and BM shall be equal when M
comes toE. For, suppose the arch ER of the circle of curva-
ture to be within EM the arch' of the curve EMH ; and> if the
angular velocity of EM about Ebe said to be less than the an-
gular velocity of BM about B at the term when M comes to £

X4 ^le

1

S^ Of the Curvakirer dfl^hut. Boot 1.

Idt it be less in the ratio of £6 to EZ^ and (by art. W&i
itsball be equal to the angular velocity of ZM about Z at that
tejfiQ^ and therefore less than the velocity of &M abput b at tbe
s^me tenziji b beiag any point betwixt B and Z. Upon £6 de-
scribe a circle £r& that touches ET^ produce £M till it meet
this circle in r Join br and 6M $ and^ since the arch £M is with-
in this circle Erfe (art. 364), ihe angle MET, or E6r, is always
gipeater than E6M while M describes EM : therefore the angvH
laa' velocity of EM about E cannot be less than the angular
velocity of frM about 6 when M comes to E ; but it was sup-
posed equal to tbe angular velocity of ZM about Z at that
term; and these are contradictory. In the same manner, it is
stiown, that the angular velocity of EM about E is not great-
ei; than the angular velr)city of B?ii about B when M comes to
E, and that these velocities are equal likewise when EM is ^ith-
ii^ the circle of curvatiu'e Ell.

389. The various properties of the circle suggest various me->
t&ods of determining the circle of curvature. The two follow-
^^S propositions arc deduced from the properties of this circle
that were described in the last article, and sometimes give more
simple constructions for determin'mg the ray of cmvature than
tbe preceding propositions*

PROP. XXXIV.

Let S (fig. 169) bt any given point in the plane of the curve BL,
-and SP be always perpendicular from it on LP the tangent

• o/'BL ; let Sy perpendicular to SL meet LG the ray of cur^
vahtre in V: and LC shall be to LV as the angular velocity

• ef 6L about Sis to the angular velocity of SP«

• » •

, L^t CI perpendicular to SL meet it in I, and, siace the aagle

LCI is equal to SLP, it follows, from what was showa in prop.
18> that when L describes the curve BL, the angalar velocity
of CL about C is equal to the angular velocity of IL about I^
which is to die. angular velocity of SL abost S as SL is. to

IL

IL (by ait. 209)^ or as LV to LC. Bat the angujtar .velocity,
of CL about C is eqnaJ to the angular velocity pf SP about S,
by art. 368* Therdbre LC is to LV as tiie angular velocity
of SL flibout S is to the amgular velocity of SP about S, or^ SA
being a right line given in poaitioa; as the flnadon of the an-
gle ASL is to the fluxbn of the angle ASP. This may be de-
monstrated likewise from art. 584^ wheie it Was shown, that-
I^I is to SP as the flujoonrof SL is to the fluxion df«SP ; from
which it follows, that LI is to LS (or LC to«LV) in the ratio
compounded of that of the fluxion of SL to the fluxion of SP,
and that of SP to SL : that is (by art 202,^08, 211, and 212},
a^ the fluxion of the angle ASL to the fluxionof ASP. >

390. Cor. I. Let LI be bisected in Q, and PM be the ray of
the circle of the ssfaie curvature with DP (the line in which P
h always found) at P ; then shall SQ, SL, and 2PM' be in coq^
tinned proportion. For, let PT be the tangent of DP at F, and
ST be always perpendicular to PT, let PM -meet LC in K;
then, because the angles SPT,SLP are equal (art. 208andS09),
and KPT, KLP are right, the angles KPS, KLS are eqnal; ft
circle passes through the four points K, L, P, S> and* the.atigle
KSP i^ right. Theiefbre, by this proposition, -MP is to KP,
or SL, AS the angular velocity of SP is to the angular velocity
of ST, or as the fluxion of the angle ASP to the fluxion of
2ASP — ASL, because the angles PST, LSP are* always ^ecjual.
But SL is to LI (or 2LQ) as the fluxion of ASP is to the flux-
ion of ASL. Therefore PM is to SL as SL is to 2SQ, or SQ,
SLy and 2PM are in continued proportion.

391 • Cor. U. When the angular velocity of SL is to the
angular velocity of SP in any invariable ratio, LC is to LV, or
U to LS, in the same invariable ratio. Thus, when AL is a
common parabola, and A is' the vertex, S the focus, the angle
ASL is double of ASP, and LI is dbub]e of US ; whidi agreesr
with what was shown in art. 3? 1 . When AL C%;'1 70)is the loga-
riihmic spiral, and Sis the centre of the spiral/ theiangular velo-
city of SP is equal to the angular velocity of SL, because the
angle ISP is invariable ; therefore, in this figure; C coincides
with V, or I witbS ; and a perpendicular to SL at S intersects
LCpecpendicolar tothe curveiffthe centreof curvature. Whfen

AL

SSO Of tic Curvature o/Liiui, Book I.

AL(fig. 169) n to equilateral hyperbola^ theangular veledtj of
ihetangeiU at L,or of the perpeildicuIarSP^iseqiial totbe angular
iRelocity of SL (by what was tbown at the end of art. SIS),
and LC is equal to LV^ or LI to LS ; but they are on oppoMte
sides of the point L. When the angular velocity of SL is ta
the angular velocity of SP in an invariaUe ratio, the angalar
vrioclties of SP and ST are likewise in an invariable ratio;
^ and PM b to PK^ or SL^ in the same ratib. For example,
vben AL is. a circle^ and S is in the circumference, PM is two
ihiids of SL. In thjacase P is in an epicycloid that is deicrib-
0I fagr a point in the circumference of a circle while it levdves
on dn equal circle ; and M is in an epicycknd of the same kind.
d^Cor.IIL I^t A£C%. 171)bearightlinegiveninposition^
8A Ik peipendicular to it in A from the given point S ; add let
10^ othar right line from S meet it in M ; let the angle ASL
be ;Jways to the angle ASM in any invariable ratio expressed
Iry that cf if to ai|it ; and> SA^ SM being the two first terms
of a j;eometi!ical progression, let SL be equal to the term of this
ffOgiression whcM pkicein the series is denoted by n-f-l.; or,
wacxt generally, lelSL be to SA as the power of SM, whose
exponent is any positifte number n, is to the same power of SA :
Aen the angle SLP (contained by SL and the tang^it at L)
rfMllbeeqpdto^tbe.angleSMA; the ray of curvatm^ LC shall
he to LV as fi is to n — 1 ; and, if SB be to SA as n — 1 is to
a-|*^> ^ variation of curvature at L shall be as AM directly,
and as SB inversdy. For, let circles described from the centre
S through M and Lmeet SA in F and/; and, the points F,/
lemainiag fcied while M and L are supposed to proceed in the
Knes AM and AL, the fluxion of the arch FM shall be to the
flujdoa of /L in the ratio compounded of that of SM to SL,
and that of the fluxion of the angle ASM to the fluxion of
ASL (or of unit to »> ; the flqxion of SM is to the fluxion of
SL in the same ratio, by art. l67j consequently, the fluxion
of SL is to the fluxion of SM as the fluxion of tlie arcb/X
i& to the fluxion of FM. Therefore the angle SLP is equal to
SM A> bjf prop. 16; and if SP be perpendicular to LP, the angle
ASL shall be to ASP in the invariable ratio of n to n — 1.
The fluxion of the angle ASL is to the fluxion of ASF, and

(by

Chap. XI* ond its Fariaiion. 331

(by this projj^itioD) the ray of curvattuw LC is to LV^ in the
same ratio. The iliuuoo of the ray of curvature LC h to th«
^uxiou of LV as n is to n — 1 (art. £4), the fluxion of IN is
to the fluxion of SL in the ratio compounded of thai of
94^1 to n and thfit of LV to SL (because LV is to SL as

SM is to SAj or as the power of SL whose exponent is l -]- *-

is to the same power of SA) ; and the fluxion of SL is to the
fluxion of the curve AL as LP is bo SL^ or AM to SM. Tliere-
fbre the fluxion of the ray of curvature LC is to the fluxion of
the cun'e AL as AM is to SB ; and the variation of curvature
(as it is understood by Sir I$aac Newton, and was exphiined in
art. 369) is as AM directly^ and SB inversely. Hence^ the
▼ariation of curvature at L in any given figure of this kind is
as the tangent of the angle contained by SL and the perpen-
dicuhur to the curve, and in any of those figures is to the vari-
ation at the point in the parabola where the right line from the
focus intersects the curve in an angle equal to SLP, as SA is
to SSB. In the logarithmic spiral AL (fg. 170) the fluxion of
the ray of curvature is to the fluxion of the curve as SC is to
SL^ or LP is to SP ; and, consequently, in the invariable ratio
of the tangent of the angle LSP to the radius. Therefore the
variation of curvature in this figure is invariable (as appears
likewise from art. 370), and is to the variation of curvature at
the point of a parabola where the right line from the focus in-
tersects the curve in an angle equal to the given angle SLP as
1 is to 3.

39s. Con IV. If we substitute a semicircle AMS (Jg. 172) iu
the place of the right line A£ in the construction of the last
article, it will appear in the same manner that the angle LSP
is equal to ASM (SP being now the same way from SL that
SM is from SA), that the ray of curvature LC is to LV as n is
to n-|-l, and that (SM being produced till it meet Am per-
pendicular to SA, in m and SB being taken to SA as n-f-l is
to n — 1) the variation of curvature at any point Lis as Ai»
directly, and SB inversely, and in any giveu figure of this kind
is as Am, or thetangent of the angle contained by SL and the
perpendicular to the ciirve.

394. Cor

1

334 Of tlu Curvature of Lines, Book I.

Sg4. Cor. V. If any curve AM (fig. 173) besubstiiutedfbr the
right line AM ift cof . 3, the Ife^t of th'e construction remaining^
it will appear in like manner that the angle SLP contained by
SL and LP th'e tangent of AL shall be equal to the angle SMT
contained by SM and the tangent ofAM. Let Mc, LC be the
ray9 of curvature at M and L^ respectirely;. let Sii, SV
perpendicalar to SM and SLmeet those rays in u and V ; and
let £r be to cu as unit is to n: then LC shall be to LV as Mc is
to Mr. The demonstration is easily deduced from the equality of
the^nglesSLP^SMT^by thisproposition. Thus^if AM beaconic
section^ S the centre, SA half the transverse axis^the angle ASL
equal to 2ASM, and SA^ SM, SL, be in continued proportion ;
t^ep AL ^all be a conic section, that shall have its focus in S»
an(^ cu being bisected in r, LC shall be to LV as Mc is to Mr.

395. Cor. VL Let CP and SP (fig. 123) revolve about the
polf s C and S with any angular velocities that are to each.
qdier in the invasiaB^e ratioof ST to CT, as in art. 315 and 316,
make the angle CPN equal to SPT, and TP/ equal to TPS, and
let P/ meet CS in L; let PO perpendicular .to PN meet CV
perpendicular to CP iii V, and let PG be to PV in the invari-
able ratio of ST to CT : then, if PO be the ray of curvature of
the line described by P, PO shall be to PG as TL is to the
sum or difference of TL and ST according to the different posi*
tion&of the points T and L with respect to the poles C and S.
The demonstration is deduced from prop. 18, and art. 315, by
this proposition. If SMC be an arch of a circle, the arch SZ
ba to SM in any given ratio, and CZ always meet the right
line SM in P, the tangent of the line described by the point
P is determined by art. 3 1 5, and its curvature by this con-
struction; because the angular velocity of CP is to the angular
velocity of CM, or of SP, in the same given ratio.

396. The following proposition is the sequel of the Idth and
£6tb^ and isi of use in enquires concerning the curvafure of
lines that are described by means of right lines revolving about
given poles,or of angles that either revolve about such poles /)r
are carried along fixed lines, in the manlier explained by several
examples in the preceding chapter from art. 315 to art. 324.

PROP.

PlatelDLflmagJg^- 1

f '

Chap. XI. dvd i7« Variation. 338

PROP. xxxr.

ir<'^ &bea given poifU in the plane of the curve II EM Qig* 174),

join EM and SM, a«d a;AeM the point "M, «» describing the

curve MEH, comes ^o E, /e^ Me angular velocity of SM.

about S 6e ^o rtc angular velocity of EM oioM^ E as ET

15 ^0 ST, and the point T be taken upon SE produced, or

. betxveen S a/id E, according as the angular motions ofSVi
and EM at that term have the same or contrary directions;
then, if ST, SE, and SB be in continued proportion, and B
be taken upon SE on the same side of the point S tDlth T, the

circle of curvature at E shall pass through B.

*

1 For, since SB is to SE as SE is to ST, BE is to SE as ET is
to ST, or as the angular velocity of SM about S to the angu-
lar velocity of EM about E at the term when M comes to E,
by the supposition. Join BM, and the angular velocity of
SM about S shall be to the angular velocity of BM about B,
yirhen M comes to E, as BE is to SE^ by art. £0g. Therefore,
the angnlar velocity of EM about E is equal to the angular
velocity of BM about B at that terjn ; and, by art. 388, the
circle of the same curvature with EM at E passes through B*
The concavity or convexity of the arch EM is towards S,. ac-
cording as the angular motions of SM and BM have the same
9r contraiy directions while M comes to E in describing ME.
, 397.. ^^* I- l^t ^^ invariable angles DEG,K^H (fig. 175),
fevolve about the poles E and S after the same manner, as in art.
319. Let N the intersection of SK and ED m.ove in any line
Yf, and M the intersection of SH apd Ej3 describe the curve

, el N come to w,EG to E^,
wd let the right line An touch I^in.n ; make the angle EnT
j^ual to SnA the contrary way from Ef that SnA is from Sn i
take SB from % towards T u|>on ST, a third proportional to
^T and SE, and a circle described upon the chord EB so as to

. touch

334 Of Jhe Curvature of Lines, SookL

touch ¥g shall be the circle of the same curvature with EM at
£. The curvature at S is determined in the same manner ;
and^ if SK touch the curve Ffatu when EG coincides with
ES, then S shall be a point of contrary flexure in the curve
EMS ; and, if Ff has a continued curvature at u, the curva-
ture of EMS at S shall be less than that of any circle* When
the right line S» is itself the tangent of F^ at «, and F/bas a
continued curvature at n, the point E is a cuspid^ and the cur-
vature at E is greater than in any circle.

398. Cor. IL Suppose that it is required to describe a cofiic
section through the points S and M 0^. 176) that shall toudi
a given right line Eg at E so that the circle of curvature at E
ahall meet SE in B. Let SA be taken from S the same way
with SB, so that SB, SE, and SA may be in continued pro-
portion ; let the angle SE^ revolve about E ; and, when the
side Eg comes to EM, and ES to Ed, let SM intersect Ecf in
N, join AN; let the right line MN revolve about the point S,
and its intersection with the side EN move Over the right Kne
ANi then shall its mtersection with the other side EM describe
the conic section required.

599. Cor. Ill (fig. 177). In the same manner, if it Is required
to describe a line of the tliird order through the double point
S, the three points M, C, and K so as to touch Eg in E, and
EB the chord of the circle of curvature at E be given; let the
point A be determined as in the last problem ; let the angI<J
SEg revolve about E, and a right line MN about S, in the ssLxne
manner; find three points N, c, and k from M, C, and K, as
N was found from M in the last problem, describe a conic
section through S, A, N, c, and k ; and, if the intersection N
move always in this conic section, M shall describe th6 line of
the third order required. If it be requured that the line shall
have a point of contrary flexure at E, the conic section is to
he described through N, c, and A so as to touch SE at S.

400. Cor. IV. The five points A, B, C, S, and E (fig. 1S6) in »
conic section being given,let it be required to determine thecu"-
rleofcui-vatureatC. Determinethe pointsD,}^n, andthctan-
gent Cn, as in art. 324, make the nngleDjr equal to SqA the
contrary way from Dq that qA is from Sgr, and let qr meet oV
in r; make the angle Dnx equal to CwA with the like precaa-

tlOIb

Chap. XL und ih Faiation. SSi

Hon, and let nx meet CD in x ; join rx, and let it meet CS ia
T ; let ST> SO, and SB be in contiBved proportion, a^d llie
. ppints B and T be on die same sideof S : then a circle describ-
ed upon the chord CB so as to touch Oi shall be the circle of
curvature at C. There arise varioiis constnietions for detemio-
ing t(be curvature of lines of the higher orders from thk {HO-
.position^ analogous to those by which the tai^ents and asymp-
totes of lines were determined in the last chapter. But tn-
. stead of insisting on these, it will be more worth while to add
here a property of the lines of the third order to those that have
been observed by Sir Isaac Newton, Envmir. linear * iertii or^
dinis,

401. LetA(>|f. 176)beanypo]ntinaIineofthe third orderfrani

which two tangents, AC, AS can be drawn to the curve in C

and S ; from any point in the curve, as P, draw riglat lines to

C and S that meet the curve again in M and N, respectively;

join CN and SM, and the intersection of these linesshall be dt

ways in the curve. If CN and SM be parallel to each other^

they shall be parallel to the asymptote of an hyperbolic branch,

or shall show the position to which the tangent of a jilkuraboltc

branch continually approaches while th^ figure is produced ;

that is, acccprding to the usual style, they shall intersect eacli

other in the curve eitlier at a finite or at an infinite distances

and if two other tangents Ac, A/ can be drawn from the samse

point A to the curve in c, and^^ the right lines that join anyof

the points of cojQtact C, S, c and/ shall intersect each othorim

thecurve. Wemay explain this property, with its consequence^

more fully on another occasion, and show how a line of the

third order (whether it have a double point or not) can be

described through seven points so as to touch two right lines

given in position at two of those points ; and shall only observe Air*

ther, to illustrate this property, that it holds of any conic section.'

and right line described in the same plane : for if CA (/^. 179)

and SA touch a conic section in CandS, and from aoy point P the

right lines PC, PS be drawn that meet the conic secsion in M

and N; then CN and SM shall always intersect each other ia

some point of the right line AP. Of this see Mn Simsou^s

Sect, conic. lib.5,propA5, Butto return to thecurvirtureoflines.

402. A

A

S36 Of tit Evolution of lanes. Bode L

402. Aflexibleliiie or ihreadlCaA (fig- 1 80)beiiig applied along
the convexity of thecurve ICafrom Ito a^ let the partaAbe ex-
tended m a right line that touches the cmve in o ; ih^i^ while
one extremity oftheline^ or thready remams fixed at I^ let the
•cjther extremity A move towards H, so that the line may be
gradnally separated from the curve^ and the part CE which is>
dot applied to.it be always extended in a right Une that touches
the curve ; then^tbe point £ sliall in this manner trace a curve
^ AEM^thatby theexcelleirtMr« Huygensig said to be describ-
ed by the evolution of a€l, which is itseif ^called the tvoluia^

PROP. XXXVI.

i.et the right line CIL tovch the evoluta aCl inC, and meet the
curve AEM, thdt w de$cribed by the eT^olution ofaCI, in E ;
then a circle ERB described Jrom the centre C through the
point E shall ha'CeVie same curvature with AEM at E : and
ifc be the centre cfthe cifck oftlu same curvature with the
evoluta aCl at C, the variation of the curvature of AEH al
E shall be rneasufed bj/ the tangent of the angle CEc*

• Let the right line QM touch the evoluta at any point Q be-
twixt C and I, arid meet the curve AEH, the circle ERB^ and
Tight fine EB in M/R, and N ; join CR, and, since the right
^ine QM is equal to the arch QC and right Une CE (or CR)
'taken together, QM is greater than QR ; consequently, the
'arch EM of the curve £MH passes vnthout the circle ERB,
.'and it is manifest that no circle described through £ with a ra-
dius less than CE can pass between EM and ER. Because the
sum of QN and NC is greater than the arch QC (art. 183),
the sum of QN and NE is greater than the sum of the arch QC
and ray CE> or QM ; therefore NE is greater than NM : and
a circle described from any point N through E, with a radius
NE greater than CE, passes without EM and ER. Therefore
no circle can pass between £M and £By and W^ ^ ^^ circle

of

risitexsJif SS8.Vo£.l

Ft^^y4Jff2.

f

^V^7S,

\

7!ffJ7j,

Chftt^.ik^L Ofih1^i>htionBf Lines. . Sst

Af the same curvature ivith tM at % by airt, S64; Jh like
tnanner it is shcnm, fhat the circle RE produced oh the other
side of E pasi^i without £A^ and that no circle can be drawd
between them on that side. It' is evident, that the curvature
decreases from E towards ll, but increases ftoih E towards A ;
Und that the curve passes withih or without thfe circle of curva-
ture, according as the curvature increases or decreases. The
Variation of curvature at E is measured by the ratio of the flux-
ion of the ray of curvature CE (or of the ctirve aC) to the
fluxion of the curve AE, or (because the angular motion of CB
tKe tangent of ntCI is e^u^ to the angular motion of cC about
c, by art. 388) by the ratio of Cc td £C, that is, by tlie tan-
gent of the angle cEC.

403. Cor, 1. The length of the arch CQ is equal to the dif-
ferrirtce of the rays of curvature MQ and EC, tvhen the cur-
vature increases or decreases continually frotn E to M. Henoei
from any geometrical curve anothet may be deduced that shall
admit of an accurate rectification.^ If we suppose (as in art,
366), that TMK is always parallefto £C, the rectangle MTK
equal to the square of ET, and BV the tangent of BKF at 6
to meetTE in V, then shall EV, EC, and one third part of Cc
the ray of curvature of the evoluta aCI, be in continued pro-
portion. For (by art. 386), when K sets out ftoih B, the flux-
ion of TK is to the fluxion of the ray of Curvature EC as 2 is
to 3 ; consequently, BE is to EV as 2Cd is to sEC : but BE
is equal to 2EC (by prop. 32), therefore EV is to EC as EC
is to one third part of Cc ; and if "kr perpendicular to the curve
BKF meet ET in r, Cc shall be equal to three fourths of Er.
Thus the curve BKP> by which the curvature of £Mli and
its variation were determined in prop. 32, and ill corollaries^
serves likewise for determining the curvature of the evoluta aCtm

404. Cor.n. Henceaready wayisdeducedforfindingthecen*'
tre of curvature of the lin6 by whos€ evolution any coni<j sec-
tion is described. Let E f^. l60)beanypointinaconic|9ectioh^
EO a diameter through E, b the centre of curvature at E; leihY
parallel to the tangent BT meet EO in Y; take 6c eqiual to SftlT
upon Yb produced fi^om 5, and c shall be the centre of carva^
ture of th^ line by tvhos^S evolution th« cbxiic $e4Ucm £MH ii

VOL. L Y dmxib^

^3S OftkeCyelMi. > BookK

described. For the kriatigles VE6, BAY cure similar (by the
5th property of the circle of curvature^ art. 375), and VE is to
E6 as £6 is to £Y, and c is the centre of curvature belonging
to the point b of the evoluta, by the last corollaiy. When E
is in the extremity of either axis, iY and consequently be va-
nish^ the evoluta has a cuspid where the curvature is such as
wasdescribedinart. S78(j^. 171). Inlikemanner,inart. 392^1et
CY parallel to the tangent LP meet 15 in Y, let Cc be to CY as
w+l is to n — 1, and c shall be the centre of curvature of
the line qCz by whose evolution LA may be described. And
"by a similar construction the centre c is determined in art. 393.
405. But to proceed now to consider some of the useful pro«
blemsthathaveadependenceon th^ curvature of lines (Jig AS I).
WheQ acirclemoves upon agiven right linesoas always to touch
this right line, and apply the parts of its circumference suoces*
sively to it, any given point in the circumference describes a
"Cycloid. Let IV be the given right line, and C the given point
in the circumference of the circle KCL touch that right line
first in I, and again in i, after describing tlie curve ICAt while
the circle makes a complete revolution; bisect li in B, and let
BA perpendicular to It meet the curve in A ; then ICAt is the
cycloid. Is its base, AB its axis, and AMB the generating circle*
If the circle proceed still in the same manner along the right
line IV, the same given point of the circumference will de-
scribe an equal and similar figure in every revolution of the ge-
nerating circle ; from which it is obvious, that this figure is not
of those that are called geometrical^ which can never meet a
right line in more than a certain definite number of points; for
It may be continued till it meet the base li, or any right line
parallel to Ithetwixt It and A&, in any assignable number of
points. It has, however, several remarkable properties useful
in philosophy, some of which we shall briefly demonstrate.
1. Any ordinate from the cycloid, as CP, perpendicular to the
axis in P that meets the semicircle AMB in M is equal to the
sum of the arch AM and of its right sine MP. For let the
generating circle KCL touch the base in K when the point that
describes the cycloid comes to C; then, since the arch KC or
BM is equal to the right line KI^ and the semicircle >BM A
. . to

Chhp.l&L Of thi Cycloid. 839

to Bl (by the' description), BK> or MC> is equal tS> die arch
CL, or AM ; and CP is equal to the sttin of the arch AM an4
its right sine MP. £. The tangent of the cycloid at C b pa^
jallel to the chord AM. For, let O be the centre of the se*
micircle AMB, join OM ; and, since the tangent of the circle
at Mis perpendicular toOM, it follows, from the fourteenth pro*
position, that the fluxion of the arch AM is to the fluxion of
AP as OM is to PM, and the fluxion of PM to Uie floxion of
AP as OP is to PM; consequently, the fluxion of CP (which is
equal to the sum of AM and PM) is to the fluxion of AP at
BP is to PM, or PM to AP : therefore (by prop. 14), the tan-
gent of the cycloid at C is parallel to the chord AM^ Si Let
CR parallel to BA meet Ab parallel to BI ia R ; and^ since the
fluxion of AR (or PC) is to the fluxion of AP as PM is to
AP, or CR, the fluxion of the area ACR is equal to the flox**
ion of the area A;rMP, by art. 111. Therefore these areas
are always equal, and the area ACI6 is equal to the semicircle
ArMBA. The right line BI is equal to the semicirCumference
AMB ; consequently, theparallelogram B& is quadruple of the
semicircle, and the area ACIB is triple of it. 4. The arch of
the cycloid AC is double of AM the chord of the arch AjM.
For the fluxion of the arch AC is to the fluxion of AP as AM
is to AP, by prop. 14^ and the second property. The right
lines AB, AM^ APare in continued proportion; oonsequentlyi
(by art. 96, or 142), the fluxion of the chord AM is to the flux-*
ion of AP as AM is to SAP. Therefore the fluxion of the
ounre AC is to the fluxion of the chord AM as 2AP is to AP^
or as 2 is to 1 ; and the arch AC is to the chord AM in the
same ratio, by art. 33 ; so that the semicycloid ACI is double
of AB the diameter of the generating circle.

406. Hence, 5, the cmre AEa described by the evolution
of the semicycloid ACI is an equal semicycloid. For, upoa .
lb produced from b take ba equal to BA,describe the semicircle
bma, let bm parallel to the chord AM meet this aemiciide in
m, and CE the tangj?nt at C meet Aft in L ; theo^ becanse the
angle 6 AM isi equal to A^m, the arch bm is equal to the, arch
AM> ai)4 the chord bm equal to the chord AM, which is equal
and parallel to CL or LE; consequently, Em i^ equal And

Ys parallel

U6 OfthcCydM. Bookt

l^rilld to AL or KI^ which is equal to the arch BM or ^nn ;
and Up the ordioiite from E peipendicular to ba is^ual to the
sum of the arditfiii> and of its right fine mp. Therefore, by the
convene of the first property^ AEa is a semicyckud that has Im
for its axisy and btna for itas gaierating semicircle. Hence the
ray of curvature of the cydoid AEa at Bis equal to £Aiii; and
the variation of curvatore is measuied by the tangent of thet
angle abm, or of the angle contained by the tangent at E and
(lie ordinate Ep> by arU 40e. It appears, likewise, that aheary
body may be made to describe any arch of a cycloid £a by
mspefiding it ftom a flexible line or thread ICE equal to 9jbA,
thai has one end fixed at I^ and is i^pUed to the convexity of
the semicydoid ICA from I to C, 1^ being peipendicslaf to
the horizon : for £a will be described by the evofcitioQ of IC :
and^ by an eqnal and similar semicydoid toudiing 16 on the
diher side at I^ a pendulum may be made to oscillate in thig

• 407* If ihemotionofthegeneraiingcirdeKCLuponthebase
1% or oi the point M in the semicircle BM A, be uniiarm> and
its vek>city be the half of that which would be genaraled by
an uniform gravity by falling through the diameter BA, then,6,
the velocity of C in describing the se&ucycloid ICA will be
equal U> tbatwhich would be acquired by the same gravity by
falling through BP, or by falling along the arch of ti^e cycloid
IC from I> BA being supposed perpendicular to the horilon.
Bor> since the arch AC is double of the chord AM, the fluxion
of the arch AM is to half the fluxion of the arch AC (or the
Ytelocily of M in the semicircle is to half the velocity of C in
the semicydoid) as AB is^ to BM (by prop. 17), or in the sub-
duplicate ratio of AB to BP ; and the vdodties aoqinrad by an
unif^Hrm gravity by falling through AB and BP are ia tbe same
ritj<^ by art. $($• Hence, the time in which a heavy body do*
steibes a densieycloid ICA> or AEa, by descending eloog it by
itt gravity, ia to the time in which it would fidi thioogii BA
tbC'dtenieter of the generating circle by the saBaegravity as the
stinfeLrctinrference of a circle is to ita diameter. For that time
i^ th|» . san^ in wluch the point K describes IB> or the point
M4escrii|tt the ftemii;0CutAference BM A» by an unMem mo*

%fm

tima

Chap. XL Of tht Cjfcloid. - 84t

tion with bdf the velocity that would be acquired by i^IUng
through BA ; and the time in which a body falls ffom B to A
by the same gravity is equal to the time in which it would de*
scribe BA with the same uniform motion^ by art. 95.

408. In general, 1, let a body descend by its gravity frpm
any point in the cycloid AEm, as H, along the curve HEsr, the
axis ba being perpendicular to the horizon ; let the ordpatf
from H meet the axis ba\nd; upon the diameter da d^cnbe
the semicircle dna ; and let Ep the ordinate from the body at
any point E always meet this semicircle in n : then shall thf
motion of the point n in the semicircle and be uniform so as to
measure the time^ and its velocity shall be equal to the half
of that which would be acquired by falling through the right
lineya a third proportional to ba and da. For the chord am
is always to the chord am in the subduplicate ratio of da to 6tf ^
and the fluxion of the chord an to the fluxion of the chord an^
(or one half of the fluxion of aE by the fourth property^ in
the same ratio. The fluxion of the arch an is to the fluxion of
the chord an (by prop. l6) as ad is to dn, or in the subduf^
cate ratio of ad to dp. Therefore the fluxion of the arch an i^
to half the fluxion of aE in the subduplicate ratio of the squaie
of ba to the rectangle contained by ba and dp, that is^ in the
Mibduplicate ratio of ya to dp ; and the velocities acquired by
falling throughyb and dp are in the same ratio. But the vel^
city in the cycloid at E is equal to that which would be acqw*
ed by falling from d through ^ ; consequently^ the velocity
of n in the semicircle dna is half of that which would be Itcquiv-
cd by falling through^. Hence, the time ih which the bo-
dy £ describes any arch of the cycloid HE is the satne in
which the point n describes the arch dn by en uniform mo-
tion with half the velocity that would be acquired by falling
through the perpendicular yb ; and the time in which the bo-
dy E descends along the arch Ha by faUing from H to a, )}
equal to the time in which the point n with th^ same umfi^noi
motion describes the semicircle dna, which is eqoal to the timf
in which thepointmdescribesthesemicirde bma uniformly with
hdf the velocity acquired by falling through ba, became dna
la tafaia as i« is to ia^<Mr in the subdupbciKke latip gtfaUn ba;

y 3 that

842 OftJic Caustics by R(flexion. Book T.

that is^ in the ratio of the velocity acquired by faUing through
fa to that acquired by falliog through ba. Therefore the time
in which a body descends along the arch Ha from any point H
to the lowermost poin^ a is equal to the time in the semicycloid
AEa, and is the same wherever the point H be in AEa from
wiiich it begins to descend^ which is the celebrated property of
the cycloid discovered by Mr. Huygens, The same property
may be demonstrated by showings that the power by which the
▼elocity of the body in the cycloid is aqcelebrated at E is always
as Ea the arch from E to the lowermost poiata; and an analogous
property is shown of the epicycloid (which is described by a giv-
en pointin the plane of acircle that revolves upon a circular base)
by Sir Isaac Newton. Produce an till it meet the circle bma
nv; andj if «e parallel to nE meet the cycloid in e, the arch
HE shall be described when the motion begins from H in the
same time that Ae is described when the motion begins from A.
409. Rays of light beiDg supposed to issue from a given pointy
and to be inflected by a given curve^ so as to make the angle of
Teflexion equal totheangIeofincidence>a curve that touches all
thereflected rays is called ihtCaustiebylUJUxion. LetS (figAS^
bethe given point from which the rays issue (which istherefore
called the focusot theincident rays),SLany incident ray^ PI^
the tangent at L, LC the ray of curvature at L, Lwi the reflect-
-ed ray constituting the angle CLmequal to CLS ; then, if the
jreflected rays always touch the curve hme, it is the caustic by
reflexion. Let SP perpendicular to the tangent LP meet it al-
ways in P a point of the curve DP ; let HME'be the curve by
the evolution of which DP is described, according to art. 402,
and let PM touch HME in M ; join SM, and produce it to m,
so that Sot be equal to 2SM : then m shall be a point in the
caustic of the curve BL, when S is the radiating point. For,
because MP is perpendicular to the curve DP, the angle MPS
js equal to the complement of SLP (by art. 211 and 212), or to
LSP, and SL is bisected by MP in K ; therefore Sm is to SM
as SL is to SK, Lm is parallel to KM the tangent of HME :
but the figure kme is similar to HME, and similarly situated,
(art. 122), therefore Lm is the tangent of hme. Because \jn
is parallel to PM, the aogle mLC is equal to MPS or LSP« er

CLS;

jChap. XI. 0/ the Canities hf R^xion. 349

CLS ; therefore^ when SL is the incident ray, Lm is the re-
flected ray, and m is a point in the caustic.

410. Let I/be taken on the reilected ray equal to IS ; and,
CR being perpendicular from the centre of curvature on this
ray in R, bisect LR in q; and qf, qR, and qm shall be in con-^
tinned proportion. For, by art. 390, y^, /L, and 2PM are in
continued proportion. Let CI be perpendicular to SL in I, and
XL be bisected in Q ; when S is on the concave side of the
curve, and LS is greater than LQ, PM is equal to the sum of
PK (or SK) and KM, and £PM is equal to the sum of/L and
Ian ; thereforey^ is to qh or jR as/L i^to Lm, or as jR is to
qm. When S is betwixt Q and L, 2PM is equal to the differ-
ence of Im and 1/, and/q is to qh as/L is to Im, or as qL (or
^R) is to qm. In like manner it appears, that when S is on the
convex side of the curve, y^, ^R (or qL), and qm are in continu-
ed proportion. When Cfig. 183) the incident ray is perpendi-
cular to the curve, the reflected ray coincides with the incident
i^ay,/with S, C with R, and qm, qC, jS are in continued pro-
portion. In general, the rectangle /qm is equal to the square
of ;R, or ^L ; and, when the incident rays are parallel, the
point m must coincide with q, and Lm be equal to one half of
LR. When LS is equal to one half of LI, and S is on the
concave side of the curve,/coincides with q, and the reflected
ray lm becomes an asymptote of the caustic.

411. LetBLCfig, 105)beacircleasinart.283,Cthecentre,CB
the radius that passesthroughS theradiating point; bisectBCin
q, let qS, qC, and ;H be in continued proportion: and, when
L comes to B, m shall come to H. ' When CS is less than one
half of CB, the curve DP has no point of contrary flexure, and
tlie caustic has no asymptote. When CS is equal to one half
of CB, the diameter through S is the asymptote of the caustic.
When CS is greater than one half of CB, but less than CB,
the caustic meets CB produced beyond B, the part of the curve
DP adjoining to B is convex towards S^ P is a point of contra-
ry flexure when SL is one fourth part of the chord LZ that
passes through S (as was shown in art. 283), and the reflected
ray is then an asymptote of the caustic.

Y 4 412, Let

|44 Q^/|Jk Causiie$ by Bsfr^ction, Book I*

41d« J^SA (fig. 189) befiright linegiven inpo^Uenjand Jjn
$hallbe to SLasthe fluxion of the angleASL is tpUieflaxion of
SASP<^4SJL Hence^ in all the curves coaslracled in art,
;394 and ^QS^ J^ iskto SL in »n invariaUe ratio^ because in
those figures the angle ABL is tp ASP in an invapable ratio ;
pnljr^ in the paraboK S being the ibcus^^ the fluxion of ASh
U eqqal to the fluxionof 2ASP| and the reflected rays 9xe paml^
lei to each other an^ to the axis of the figure.

4l3. When the rays that issue from a given point are Tcfract-
td at the corve^ so that the ^ine of the angle contained hy the
);efracte(l ray and the pe;rpen.dicnlar to the curve is always to
the sine of the angle contflined by the incident ray and that per->
^ndicular^ in one constant ratio^ the curve that touches all the
tefracted rays i^ called the Cat<i/iV:iy.ft{/rac/io;i, LetSC/f^. 184)
be the r^diatii^g pointy Sf^ any incident ray^ LR the refracted
ray^ C the centre of the curvature at I^ CI perpendicMlar fiom
C on the incident ray in Ij, CK pefpefidicular o^ the refi-acled
Tay ii^ Jl ; join SC and IR \ let RZ perpendicular to Rl^meet
CI ia Z ; join LZ meeting SC in Qj^ and let QM paralJeJ tQ
^Z meet th^ reacted ray LR in M : then shall M be in tbe
caustip. far, ^uppo&iog the refracted ray l*R to touch the eaus«
tic in any point M, and LT parallel to CI to meet SC in T*
^nd the point L in describing the curve BL to wove towards
/ ; then the angular velocity of SI about S shall be to the an^
gular x^locity of ]y{R about M in tbe ratio compounded of
yie direct ratio of the fluxion of CI to the fluxion of CR (or
the ratio of CI to CR, by art- 27, the ratio of CI the sine of
tb.9 ?Pgle of incidence to CR the sine of the angle of lefractioQ
liein^ supposed invariable), and of the inverse ratio of SI t<>
JIM- The ai^ai: velocity of SL about S 13 to the angular
velocity of ML ahout M (by art. «08 and 902) in th^ direct
ratiQ of LI to LR (or of CZ to CR, because the tribes
CZ|l, LIR are similar), and the inverse yatio of SL to LM.
Th^gCpie LM i§ to RM 1^ the rectangle contained % C( aid
SL i? to ^he r^ptai^le contained by SI and CZ, pr (becan^

SI is to SL asCI is to LT) as H is V> CZ, or LQ tp ZQ;
consequently, QM and ZI^ ai;e paralH w4 the poipt M wa%
rightly determined. When the incident fays are parallel, let
CB parallel to those rays through the centre of curvature meet

LZ

€ik$f, XL Qftk4 CamUet iy Refraction. 345

X2 in Q, ikod QM pnuradklioBZ shall meet the refracted ray LR ia
thecaiMticatM. WhenC%. 185)themcidentraybecQmesper«
pendiciilar to the refiracting curve, that is, when L comes to
S^ the aagQlar velocitj of SL is to the angular velocity of ML
Of MB i» to SB ; therefore the caustic meets SC in M, so that
CM ia to 6M io ibe ratio compounded of the given ratio of
CH (the sine of refraction) to CI (the sine of incidence), and
Qf that of SC to SB : and when the incident rays are parallel,
CM is to BM as the sine of refraction is to the sine of incidence*

414. Describe the circle UCVCfig. 1 84) upon the diameter LC,
lat C V parallel to the incident ray SI meet this circle in V ; and
JIZ shall always pass through V> the centre of curvature C
;and incident ray SL being given* Therefore the point M is de**
iermiped by applying CR in this circle so that it may be to CI
in the given ratio of the sine of refraction to the sine of inoi*
4^nee, joining VR that meets CI perpendicular from C to the
incident ray and Z, joining LZ that meets SC in Q, and drawl-
ing QM parallel to VR. When LZ is parallel to SC, the.re*
^raqted ray is the asymptote of the caustic,

415. LetBL (Jig. 186) bea circle,and, the incidentrays being
fupposed parallel, let the caustic touch the refracted ray LM in
M a point of Ihe circle ; then, because LR is equal to RM,
LQ and CI are bisected in Z : and, the triangles CRZ, LRI
)>eing similar, it follows, that CZ, or one half of CI, ia to CR
w LI is to LR. Therefore, if the invariable ratio of the sine
»f incidence to the sine of refraction be expressed by that of I
|o R^ the square of I shall be to the square of fiR as the square
ipf IJ to the square of LR, and 4RR — II shall be to II as
the difference of the squares of LR and LI, or of CI and CR^
|S to the square of LI ; consequently, SRR is to II— RR as
the square of the radius CL is to the square of LI, or CV :
and in this manner Sir Isaac Newton determines the position
f f the ray that, after a refraction at L, a reflexion at M, and a
second refraction at 6, defines the interior rainbow ; for, the
lays incident about L being refracted so as to touch the catiilio
near to M, a point in the circle BLM,and being thence reflects
/ad,theywillemergenearlyparatlelat Go^.187)- Ifitbereqiww
^edthalt the rays xefiracted aboniL should be leflected al i ia

the

546 Of centripetal Forces. BdokT

the circle nearly parallel to each other; tben^ by art. 41J>
MR must be equal to M/^ or one half of LR, ZQ must be
one half of LZ, CZ one half of Z\, and therefore equal to
one third part of CL But LI is to LR as CZ is to CR^ or
as I is to SR^ and 9RR — II is to II — RR as the square of
CI is to the square of LI ; therefore 8RR is to II — RR as
the square of the radius CL is to the square of LI^ or CV ;
.and hence the ray is determined which after a refraction at L,
two reflexions at / and 6, and a second refraction at H^ de^
fines the exterior rainbow ; for, if Gm be one fourth part of
GH, or of L/, the rays reflected at 6 shall touch a caustic
formed by this second reflexion at m, and emerge parallel after
&eir lefraction at H.

4l6. Of the various problems that depend on the curvature
of Unesj none are more useful in philosophy than those which
relate to the centripetal and centrifugal forces. In this doctrine
itis supposed^ that a body at res^t never moves of itself^ and that
a body in motion never changes the velocity or direction of its
motion of itself ; but that every motion would continue uni/brm,
and its dtrectionrectilineal,unlesssome external fmce or resist-
ance affected iu Hence^ when a body at rest always tends to
move, or when the velocity of any rectilineal motion is acce*
krated continually, or when the direction of a motion is con-
tinually varied^ and a curve line described, these are supposed to
proceed equally from the influence of some power that acts in-»
cessantly ; which may be measured either by the pressure of the
quiescent body against the obstacle that binders it to move in
the first cdse, or by the acceleration of the motion in the second,
or by the flexure of the curve described in the third case; due
regard being had to the time in which these eflfects are produced,
and the other circumstances^ according to the principles of me*
chanics. Effects of tlie power of gravity of each kind fall un*
der our constant observation near the surface of the earth ; for
the same power which renders bodies heavy while they are at
rest accelerates them when they descend perpendicularly, and
bends their motion into a curve line when they are projected in
any other direction than that of their gravity. But we have
access to judge of the powers that act on the celestial bodieg

by

*L^

Chap. XI. Of centripetal Forces. 347

by effects of the last kind only; and it is of tbe^c chiefly we
are to treat here.

417- As the velocity of a variable motion is measured hy the
space which would be described by it in a given ttme^ if it was
continued uniformly for that time (art. 4, 5, and 6), and not by
the space that is actually described by the variable motion ia
that time; so the power by which the rectilineal motion of a
given body is continually accelerated or retarded is measured
by the increment or decrement of the velocity that would be
generated by that power in a given time/ if its action^ or influ-
ence, was continued -umformly for that time, and not by the in-
crement or decrement of the velocity that is actually generated,
if the action of the power varies. The fluxioii of the velocity
is measured in the same manner as was'explained in art 70.
Therefore the power that accelerates or retards a rectilineal
motion is always measured by the fluxion of the velocity of the
motion, or (because the velocity is itself the first fluxion of
the space) by the second fluxion of the space described by the
motion, the time being supposed to flow uniformly. Thus,
when a given body ascends, or descends, in the right line that
is in the direction of its gravity, in spaces void of resistance, the
power that accelerates or retards its motion at any term of the
time (that is, the accelerating force of its gravity) is measured
by the fluxion of the velocity; and if the velocity increase or
decrease uniformly so that its fluxion be constant, the gravity
must be supposed uniform. If the body descend, or ascend, in
a medium that resists its motion, the power that accelerates or
retards its velocity (that is, the difference of the gravity and
resistance when the body descends, and their sum when it
ascends) is still measured by the fluxion of the velocity, or the
second fluxion of the space that is described by the motion, the
time btiiSff supposed to flow uniformly.

418. Suppose the right line Dd {Jig. 188) to move parallel to
2tself along the given lineAOwith an uniform motion, and the gra«
vity to act always in the direction Dd; then the body will de«
scend in this line Dd in the same manner as if the line was qui*
escent, and the gravity acted in its direction. Therefore the
graviljr will be still measured by the fluxion of the velocity

with

S48 Ofuniripetal Fonti. Book L

with wbich the body descends in the right Hne J)d, or by the
second flaxion of DE the space described by it in this line, that
is^by the second finxion of the ordinate of the curve traoed by
the body on the immovable plane AOoo, the time (or the right
line AD which flows in the same manner as the time) being sup*
posed to flow uniformly.

419. When a curve is described by a gravity that acts in pa*
rallel lines, its force is as the square of the velocity of the bo*
dy directly and that chordof the circleof curvature which passes
thiough the body in the direction of the gravity inversdy.
For, the same things being supposed as in the last art. let the
velocity of the point D or the fluxion of AD be represented by
a given line DG, let GH parallel to the ordinate DE meet the
Ungent ET in T ; and the fluxion of the curve FE, or the ve^
lodty o( the body in describing it, shall be measured by ET,
by prop. 14. Let the circle of curvature at E meet Ed in Bj
and the second fluxion of DE shall be measured by a right line
that is a third proportional to | EBand ET, by prop. S3. There-
fore the gravity, which is measured by the second fluxion of
DE, is as the square of ET directly, and EB inversely, that is^
as the square of the velocity directly, and the chordof the circle
of curvature which passes through E in the direction of the gra-
vity inversely.

420. This may be further illustrated, if it seem necessary, in
the following manner. If the gravity act uniformly and in pa-
rallel lines upon a body that sets out from £ in the direction ET«
and bend its course into the curve ER,and the right lines TR,
Vor in the direction of the gravity meet ET in T and V^ and
meet the curve in R and x, \x shall be to TR as the square of
EY is to the square of ET, by what was shown in art. 95; con*
«equentiy,the rectangle contained by Vx and an invariable right
line is equal to the square of EV, and the curve EjtR » a parar
bohu And it may be shown, conversely, from art. 254, that
when a parabola ErR is described by a gravity that actsalways
in a direction parallel to the axis of the figure, then, DG being
given^ TR is invariable ; andj consequently, the gravity is uiu-
form. Let HEA be any curve line that touches the parabola
ExR, aQd h9s the same c\ir?atuiewithit atE; then^if the power

by

dup. XL Ofcentnpelat Forut. 349

by which EH id described act always in the same direction as
tfaft uniform granty by which the parabola ER isdescribed^ and
the velocities at E in EH and £R be eqnal, the gravity in the
eimfe AEH at E must be equal to the uniform gravity in the pa-
rabola ER« For if the gravity in EH at E be said to be greater
than the uniibmi gravity in the parabola ER^ let it be greater
in the ratio of TZ to TR ; and let EZbe a parabola described
through E and Z that has the same tangent ET and diameter
Ed with the parabola ER at E. Then^ the gravity in the curve
£H at E shall be equal to the uniform gravity in the parabola
£Z : and> if the variable gravity in EH be supposed first to
increase while the body moves from E to H^ TZ shall be always
iosa than TH^ because a space described by a power that acts
UttiibnAly must be less than the space which is described in
die same time when that power continually increases from thft
begfidnidg of the time. But TR is always jess than TZ ; theie^
Dmm the parabola EZ posses between the curve EH and para^
bola Br. But EH and ER were supposed to have the same
<n2rvature 'zi E ; consequently^ no parabola can pass between
them^ by art. 371* And these being contradictory^ it follows^
that^ when the gravity in EH increases from E to H^ the gra*'
▼ity at & in EH cannot be greater than the uniform gravity in
th^ ^Hbok ER. if the gravity in EH be supposedto decrease
fiwnB to Hi let a parabola Eu be described through any point »
belweeti R and Z> so as to have the same tangent and diameter
at E with ER and EZ ; then, since the gravitjr in EH at fi iift
equal t6 the uniform gravity in the parabola EZ (by the suppo*
tili<m)/ il must be greater than the uniform gravity in the para*
bolaEti^which is greater than the uniform gravity in ER. There^
fore the part of the parabola Et< that is adjoining to E passes
l>etween the curve EH and parabola ER; and (by art. 371)
BR has not the same curvature with EH^ against the supposi^
tion. In like manner it may be shown^ that the gravity in EH
kt E is not less than the uniform gravity in the parabola ER.
Therefore they are equal to each other. Let the conmum
circle of t^urvatuie meet £^ in B, and TR^ ET, and EB shal) be
in continued proportion, by art. 371, and TR is as the square
M ET directly, and EB inversely. Therefore the gravity in EH

At

1

350 Of centripttut Forces. Book H

at E is as the square of the velocity at E (which is measured by
ET, DG being given) directly^ apd as the chord of tiie circle
of carvature through E in the direction of the gravity in«-
versely ; for the uoiform gravity in the parabola £R is mea^
raced by STR, by art. 74 and 75.

4£1 . When the Telocity and direction of the motion and the
foxsce and direction of the gravity at £ are given^ the curvature
at £ is given. For^ let £T be the space that would be described
in a given time by the motion at E continued uniformly > let
TR be equal to the space described in the same time by a body
faUing from E in the right line Ed when the gravity at £ k
coptiuued uniformly; then EB the chord of the circle of cur-
vature shall be a third proportional to TR and ET ; and^ since
the angle BET is given^ the centre and ray of curvature are
given. The curvature therefore of EH at £ depends only on
the force and direction of the gravity at E^ the velocity and
dii?ection of the projection at E being given, and not on the sub-
sequent variations of the gravity. When the force i^nd direc-
tion of the gravity with the velocity at £ are given, EBis
determined, and tlie ray of curvature is reciprocaliy as the
siiie of the angle BET.

422. Let £H be now described by a centripetal force di-
rected towards a given poin t S in the right line EB ; and let £R
be the parabola that would be described by the same centripe-
tal force at E continued uniformly in a direction always pa-
rallel to EB : then the contact of the curve EH (that is de-
scribed with a variable centripetal force directed towards S)
with the parabola ER (that is described with the centripetal
force in EH at E continued uniformly from that term) shall
be closer than the contact of EH with a parabola that is de-
scribed by a greater or less foro^ continued uniformly in the
same direction. Therefore the curve EH and parabola £R
have the same curvature at £• From which it follows, that
the centripetal force in the curve EH at £ directed towards
any point S is as the square of the velocity of the motion at £
directly, and EB the chord of the circle of curvature that
passes through S the centre of the forces inversely.

423. Let SE be produced till it meet the right line AD gireu
in position \ and, if AD be supposed to flow uniformly, the cen-
tripetal

'P\ate:xajPa.a^.VoL I

V^.J^M^/

Chftp.'XI. Of cenfriptfal Fofce%. S5I

tripetal fdrce shall be measured by the second fluxbh of the
ordinate D£^ if the centre S to which this force is directed be
any where in the right line EB produced from £ on the concave
aide of the curve; because the second fluxion of DEconsidered
as the ordinate from the curve EH is equal to the second fluz«-
ion of DEconsidered as the ordinateirom the parabola ER that
has the same curvature with EH at E, by art. 383. In the sam^
ttiann^ it appears, that^ if the curve £H be described by a cen*
trifiigal fprce directed to aoy point in the right line EB pro-
duced from E on the convex side of the curve, this force shall
be likewise measured by the secondfluxion of thesameordinateL
424. Since the square of £T (fig. 1 89) is equal to the rectangle
contained by Til and EB, it follows, that tlie velocities at any
two points of curve Unes are to each other in the ratio com«-
pounded of the subduplicate ratio of the centripetal forces at
these points, and the subduplicate ratioof the chords of the cir-
cles of curvature which pass through these points and the cen*
tres of the farces. When the gravity is given, the velocities are
in the subduplicate ratio of these chords. Therefore the velocity
at any point £ is to the velocity of a body that describes a cir-
cle about S in a point at the same distance SE by the same cen-
tripetal force in the subduplicate ratio of EB to 2SE, because
the chord of the circle that passes through the centre is its
diameter, and is equal to 2SE; consequently, if SP be always
perpendicular from S on the tangent ET, the velocity in
the curve EH at £ shall be to the velocity in such a cir*
cle in the subduplicate ratio of the angular velocity of SE
about S to the angular velocity of SP aboutS ; because, by arC
389, these angular velocities are to each other in the ratio of
EB to 2SE. Tlius, for example, the velocity in a parabola,
when it is described by a centripetal force directed towards the
focus, is to the velocity in a circle that is described by the same
centripetal force at the same distance in the subduplicate
ratio of £ to 1 ; because, if A be the vertex of the parabola,
the angle ASE is always double of ASP. In the logarithmic
spiral these velocities are always equal, because the angular
velocities of SE and SP are equal in it, by art. 349. In all thai
figures constructed in art. 3^9, and 393, these velocities are to
each other in an invariable ratio when the centripetal force is

directed

358 Cf ceRtripttal Fared. BookiL

directed towards S (fig. 171 and 172), viz. in the sabduplicate
ratio of n to n — 1 in those of art 392^ and in the fiobdnplicate
ratio of n to fi+1 in thpse of art. S9dj bcpatiAC the angular
wlocities of SL and SP about S are to each other in the saAie
invariable ratio as the angles ASL» ASP in those figures*

.425 (fSg. 158). When a conic section is described by a ceii'-
tripetal force directed towards any point S in the plane ^ tii^
aecttOD, let Oa the semidiameter paraUel to the tangent at B
meet SE in R ,- and the velocity in the section at E shall be
lo the velocity in a circle described in a line at the same dia^
tOBoe SE by the same centripetal force as 0« is toatnean pro-
portional betwixt SE and RE. For, by the foordi property
of the circle of curvature demonstrated in art. 375^ the reci*
^ngle RE& (£i being one half of £B) is equal to the aqwoe
of Oa^ and EB is to 2SE as the square of Oa is to therectang^
RES. When the centre of the forces coincides iiith O the
centre of the figure, the velocity in the conic sectioB it to th^
irdocity in the circle as Oa to OE ; because, in this case, StL
vtid RE coincide with OE. When S is in the focus of dif^ sec*
tion^ these vdodUes are to each other in the ratio of Oa to a
mean proportional between the distance SE and half the trana*-
Yerse axis of the figure (which in this case lH eqbal to RB)^ ot
in the subdaplicale ratio of the distance of the revolving body
from the other focus to half the tratMverse axis ; and thes^
velocities are equal at the extremity of the shorter axis in the

4d6* The velocity and direction of the motion and the cen«
tripetal force at any given point E,with ihecentreOof the come
section, towards which the force is supposed to be directed,
being given, the section is determined that can be described by
•Qch a centripetal force. For, if Tr be equal to the space
that a body failing from E in the right lineOE would describe
^r its gravity at E continued uniformly in the same time that
ET would be described by the motion in the curve at E con-
tkmed uniformly, a third proportional to Hr and £T shall be
aqoal to the parameter of the diameter that passes through £;
tbavifore the .^eitlidiaf^ter Oa parallel to ET is given in pod*
tit>n:and magnitude, and the conjugate semidiameter OE being
lilonfise given> the conic section is determined*

4&7< When

.•

tSMp. XI Of teniripelal Porca. S55

4£7« When the. conic section is described by a ceptripeial
ferce that is directed towards the focus S (^g. 159), let T:i^ be
the space that would be described by a body falling from £ in
the right line £S with the gravity at E continued uniformly^ in'
the same time that ET would be described by the lAotion of the
body at EcontinQed uniformly ih the tangent; let EB be to ET *
as FT is toTx, and EQ be equal to one fourth pan of EB : then
a third proportional to SQ andSEshall be equal tothe.transverse
axis of the figure. Let SP be perpendicular from S on the tan-
gent ET in P, bisect SE ip r, join Pr, let the point O be taken
upon the right line Pr (the same or the contrary way from P
with r according as ES is greater or less than £Q)^ so that PO
may be equal to one half of the transverse axis, and O shall be
the centre of the figure, by what was shown in art. 3^5. When
£Q is equal to ES, the figure is a parabola, the vertex of which
is determined in that article; when EQ is less than ES, the
figure is an ellipse ; and when EQ is greater than ES, it is an
hyperbola. It appears, likewise, that, when .the centripetal force
and velocity at £ with the distance S£ ate given,' the transverse
axis is given, whatever ^e direction of the motion at £ (or the
angle SET) may be.

4£8. Let EH (Jig. 189) be any curve described by a centric-
petal force directed towards S ; let the force at £ be represented'
by the right line EK, and KV be perpendicular to the tangent
£T in V; then EV shall i^present the force by which the veW
city of a body is accelerated in a void'if it b projected from E to-
wards T, or retarded if it is projected towards t ; and KV shall
represent the force by which its course is injected from the
tangent ET, or £/. Because KV is to £K as Eft is to the ray of
curvature EC, and EK is as the square of the velocity at E di--'
rectly, and Eft inversely, it follows, that KVts as the square of the
velocity at E directly, and the ray ofcurvattu^ EC inversely, and
that the curvature of EH at £ depends on the velocity at Eand
the force KV only. Hence, if the force EV be inclreased or di-
minished by any new force that acts in the direction of the
tangent, and is in any assignable ratio to the force EK, but the
force KV remain, the curvature at E of the arch EH described

by the body shall remain the same as if that new force had not
VOL* L Z ' acted.

SS4 Of tcntnpttdl torch. Book C

acted, tf we stippose t^e same curve . to be describea by a cen-
^ipetal/orce directed towards any other pointy^ and tbe velo-
city at B to ^ tbe'saoie as formerly^ let Ki and S/ parallel to
the tangent E^meet/^ in k and /. and Cd be perpendicular to
/E m o,^ then the centripetal iorce towards/ shall be to the
centripeUlibrce that was directed towards S in the formercaae
as £j( IS to Ed (art. 4^2) ; that is^ as £^ is to £K (because the
angles i£^ is^<][ual to £^^> or as E/ is to ES.
^. 42^. It^foflows from what is shown in the last article^ ^aty
lyhen a body describes any curve '£tl by a centripetal force di-
rected towards S* in a medium tlie resistance of which is in an
asip^nable ratio to the centripetal force^ thu force is still at £
i|8 the square of the velocity directly^ and the chord of toe circle
of c^trv^ture that passes through § inverely. For the resistance
a^ects th? force EV on^y^ which is in the direction of the tan-
gent or of the mo^on of the body^ and ftieveldcitym the curve
is^celerated pr retarded in. this case "by the difference or sum
of the foro^ ^v and the resistance; hut ihe force KV that is
perpendicuiar to the tangent (upon which the curvature of £H
at£ depends)remam8 the same as when the motion ism a void;
and. since the curvature of EH at E is the same in the medium
as in the void >vhen the velocil^ and centiipetal force at E are
^e samcj it follows^ that the centripetal force at IS in the me-
ditim as well a$ in a void is ^ the. square of the velocity at £
directlvi and E6 inversely. The velocity in the curve at E ia
e medium is Ux the velocity in a circle described in a void by
€ saine centripetal fof ce at the same distance, in the same ratio
as if the curve was described in u void by a centripetal force
cirectod to^ the same centre S^ that is^ m the subduplicate ratio
of the anfiidw velocity of SE to the angular velocity of SP
about S^ by art. 424. Therefore what has been shown in the
preceding, articles on. this subject i^ to be extended equally to
both cases: .and for this reason we chose first to shew how the
Telocity m the curve i^ compared with the velocity in a circle
described bv the same ceptcipetal force at the same distance,
before we should enquire into the variations of the velocity iaind
oenUripi^tal^Cfce in agivea fijgur^, H^i^h S^e different in.a me-
oium from what they are in a space void of resistauCe, Thus^

if

m

Clmp.XL Of centripetal Foffef. ^

ifal^gfriUimlc ppini}^ for ^^ample>be,468crib^eKherm^a me-
dium or in a void-by a ^orce directed to the ceotre S, the velo-
culy i« aiw9jB eqyal \o the velofiiy in a cirele described in a
Void at the same distance by the same centrqpetftl forcis : and
in Qi^ of the figures constmcted in art* 392 an4 S9d> these ve-»
locities are in an invariable ratio when S is the centre of the
£>rces.

430. I^ followf^ that the centripetal forces at E directed to-,
wards a give? pQiat S by which the same curve £H can be de-
9crib^d in a qiedMm wd in a void> are to each other in the du-
plicate raUo of the velocity at £ in the medium to the veloci^
at the ^aiki^ point in a viud ; becwse the diord of curvature ia
the same in both cases.

43 K Another theorem C%. i.QS)by^hicht|iecentripetalforcet
may be dtscoveredlbllowf iTO{nwhatwas8hQwninart385. Let
£V a circle desoribed from tbepenfie S meet PM in V ; and
tbe s^Hd fluxion of FM $haU b^ equ^ to the difference of
the second Au^ons of SM and PY wh^ the points M and V
set out together from JB» Tih«refi>rej if ^re suppose that viiuh
the body M descrSNes the onrre llEiA bj a force directed to»
wards S^ another body V revolves in the circle EV^ the uni«>
form wigular velocity of Sy abcfot S ibeing equal to the angular
TdociJy of SM aboitit $ whepM sets out^om E, and that a thjrd
body Ji descends or ascends in the right line S£ so that SL is al-
ways equal toSAf ; then the centripetal force of M in the curve
HEM at EahaU be equ*il to the di^sren^e or sum of the centn*
petal force by whicli V is retained in the qirple £V> and of the
£wce by which the motion pf L is accelerated or retarded at
£ in the right line 3E. for the centripetal force in the cird^
£V is measured by the second fluxion of PV when V sets out
from B, AD being supposed to 4ow uniforn^y, by art. 423^ and
Ibe force by which the motion of L is accelerated or retarded
in the right line S£ is measured by the-sepond fluxion of SL>or
'SM> when L comes to E, hy art* 4 18. Therefore the diiSerenc^
or sum of those foroes is measured by (ihe difference of thesp
aecond fiuxicm^ of PV and SM, or the seeped fluxion of P3^
and, amaequently, is ^nal to the centripetal force in HEM at
B,. by art. 4flS. The centripetal force in the cirdje EV is equal

Za • to

556 Of centripetal Forces. • Bookl/

to the eentrifiigld force that arises from the motion of rota-
tion m the same circle^ or in the curve EM at E^ since (by
the supposition) the angular velocities of SM and SV are equal
when M and Vset out from E. And hence it appears^ that the
force which accelerates or retards the velocity of Lat E in the
right line ES (which is sometimes called the paracentric velo-
city of M at E) is equal to the di£Ference betwixt the centripe*
tal force in EM at E and the centrifugal force in the circle EV,
and not to the difference betwixt the centripetal force in EM'
at E and twice the centrifugal force in the circle EV C/ig. igo).
It follows from this^that,when two curve lines EM^ em are describ-
ed by centripetal forces directed towards S^ and SM is always
equal to Sm, the difference of these forces at E and c must be
equal to the difference of the centripetal forces in the circles
EV, et, the angular velocities of SV and Sv being respective-
ly eqnd to the angular velocities of SM and Sm when M and
m set out from E and e. And, if the velocity in the circle EV
be to the velocity in the circle fv as G is to F, the difference
of the forces in EM and em at E and e shall be to the force in
the circle EV as the difference of the squares of 6 and F is to'
the square of G.

432. When a circle is described by a centripetal force that
is directed towards its centre, the motion is uniform ; for, the
direction of the centripetal force being always perpendicular to
the tangent, it neither accelerates nor retards the velocity of the
body, and has no other efiect but to bend its course continual-
ly from the tangent into the cirde. The centripetal forces are
as the squares of the velocities directly, and the diameters of the
circles inversely, by art. 422, or (because the velocities are as
the diameters directly, and the lilnes in which the revolutions
are completed inversely) as the diameters directly, and the
8(][uares of the periodic times inversely. If rays be supposed to
be drawn always from the centres to the revolving bodies, the
centripetal forces shall be likewise in the ratio compounded of
the ratio of the angular velocities of those rays (or of the an«
gular velocities of the tangents at the bodies), and the ratio of
the velocities of the revolving bodies. Thus, when the circles
are described in equal periodic tingtes> or the angular velocities

- * of

jCfiap. XL Of centripetal Forcct. fWI

of tho$e rays are equal, the velocities and the centripetal forces
are as the distances of the revolving bodies from the centres
of the circles. When the squares of the periodic times are as
the cubes of the rays of the circles, and, consequently^ the ve-
locities in the subduplicate ratio of these rays inversely, the
centripetal forces are reciprocally as the squares of the rays.
When the periodic times are as the squares of the rays, or Uie
velocities reciprocally as the rays, the centripetal forces are re-
ciprocally as the cubes of the rays. In general, when the ve-
locities are reciprocally as any power of the rays whose expo^
nent is m, the centripetal forces are reciprocity as the power
of those rays whose exponent is Qm-^-l : and, conversely,
when the centripetal forces are reciprocally as the power of the
rays whose exponent is n, the velocities are reciprocally as the
.power of the rays whose exponent is one half ofn — 1.

433. The velocity in a circle is to the velocity that would
be acquired by falling in a right line from the circumference to
the centre by the same centripetal force with which the circle
is described continued unifcurmly, as the radius is to the side of
the inscribed square, or as unit is to the square root of two,
LetSbe the centre, SA (fig. 19 1) the radiusof the circle^ and sup.
pose thecentripetal force atAto continue to act upon thebodyuni-
Ibrmly and in parallel lines ; then shall it describe aparabola AR
that has the same curvature with the circle at A, or that has its
focus in the point F,if SAbe bisected in F(by a^. 429 and 371).
liet 6 A be equal to AF, and it is known that the velocity in this
parabola at A is equal to the velocity that would be acquired by
iUlingwith the sameuniform centripetal force from 6 to A; and
therefore is to the velocity that would be acquired by falling
from A to S with the-same centripetal force in the subduplicate
ratio of G A to AS (art. 95), or of 1 to 2 ; and, since the velo->
city in the circle is supposed equal to the velocity in the parabo-
la in A, it must be to the velocity acquired by falling from A
to S in the same ratio of 1 to the square root of 2. Hence the
periodic time in a circle is to the time in which a body would
fall directly from the circumference to tUe centre by the same
gravity, as the circumference of the circle is to the side of the
iiiscribed square. In general, it appears in the same manner,

Z 3 that

Sm i)/ cekifhctal threes. . -Bookie

that the velocity in any curve at E is equal to iKe vetodly that
would be acquired in a void by falling directly through one
Jburth part of the chord of the cifcleW curvatufe EB, by the
gravity at E continued uniformly ; or that Eft is always double
of the space in falling through which the velocity at E would
be generated in a void by the gravity at E continued uniformly.
434. Itwasshowninart.g5 0^g.21)that^whenamotionisnni-
formly accelerated^ that is^ when the force that accderates it is
constantyif adbe the space described by the motion^ and the tri«
angular area ADE be always equal to the rectangle ae contain^
edhv €ul and the iuvwable right hneqfy the time of the mo«
lion shall be represented by the base AD^ iiifkl the velocity by
I)£. ^he force that accelerates the motion isras IH the fltut-*
ion of the velocity directly^ and £f the fluxion'Sf the time in*
VjBrsely (by art. 114or4I7)» and therefore ihay be represent*
1^ by a jight line ixft that is to the invariable right line ijf as
*IH is to EI, or pE' to AlJ, or as the square of DE is to tt»
rectangle ADE which is equal to twice the rectangleySrcf; con-
sequently, the'squareof DE.V^hich measures the velocity at t}
'is equal to twice^bie rectangle bad contained by ha Vvfaich mea^
' sures the c^ntrip^ial force^ arid ad the space described from tfie
beginning of the motion ; and the fluxion of the square of the
velocity is measured by the double rectungle contained by ia
>nd ihe iright Une which measures the fluxion of the space,
' Hence|ifabodybesupposedtofaDintherightline«S(^^. f91)by
^centripetal farce in the direction (tSthat atahycBstanceSQis
/measured by the ordinate QN^the fluxibnof the square of the
yelbcity at Q shall be measured by twice the rectangle contain-
ed byQN and the right line which measures the fluxion of aQ :
but this rectangle measures the fluxioiTof the areaizQNd^ by
prop,' 3, therefore the square of the velocity acquired by fkllihg
^ from a to Q is measured hy twice the area dQffd, by art 24.
Xbis^ if it was necessary^ might be demonstrated in the same
fanner i^ the third proposition^ from this principle, which- is
'^ analojjous to the first axiom, that'when amotion is accelerated
by acVntripetal force that increases continually, the velocityge-
. nmted by it while a given space is describietl, is greater than
the velocity which would have been generated if the accelerat-
ing

t

4C)aa]j« XL . Of c^irijpt^al Forces. 559

ips; force had not increased^ but coi\|inued unifonn. It is like-
;wise obvious, that if aQ was a column of an homc^heoiis
fluid, imd the gravitation of any particle at the distance SQ w^
represented hy the ordinate QN/t|he pressure of^the fluid atQ
lyould l^e as the area aQfidi or as the square of (he vel^ify
that would be acquu;ed by falling from a to Q. ' * '

435. Let the velocity m the curve AE.at A. bfc equal to that
.^hich would be acquired by falling nrom a to A ih 'the ng&t
line aS. so that its square may be represented by twice Ihe drea
aADa; kt S£ be equal to SQ, and tne centripetal force iat £
t)e supposed equal tp the centripetal force at Q wnich is repre-
^^n\eA by the, ordinate QNj let JSK be equal' to ON; and KV
)>e .perpendicular to the tangent ,tij^ in V : then the force fiy
jvhich the motion in.the cui-ve at.E w accelerated snaH ffe'tfe-
!^f^se»ted by EV, and the fluxion of the sqiiare of ihe vefocity
at E by. twice the rectangle contained by EV and ^h'e right line
;^hi9h measures, t,he fluxion of the cuirvie A'&j'hj the last ar-
ticle : . but this rectangle is equal to the rectangle contained by
EK. or QN. and the riffht line which ineasuires the fluxion of
aQ, ; ^)eci^se (by^ prop. , 17) the fluxion of the curve AE is to
tfift fluxion of aQ as EK is to EV" ; therefore the fluxion of tiie
square of the velocity at E is measured by twice the fluxion of
Jthe area aQNrf; afid the square of the velocity at ^ is measiir-
^t^d by.2aQN^. The square of the samie velocity is measured

^ by t;wice the rectangle contain^^ %'^^ ^^4 1^ 0>y ^* 433),

, or^by tbejepfangle contained by QN and E6 ; consequently^ tne

xeciapgle contained by QN. an^ ^b is double of the kxe&aQl9d:

and hence, when the centripetal force at *£. or Q. is khown.

and the l^w according to which this force varies in diflerent dis-

.taPjCes, .E6 i^founJby thequaaratureofthea^^ *«Let

.the, ?fluare of tne rig]it line jQR be equal to twice the iarea

aQSdj, that QR may represent the velocity at E; then QN,

. QR, and p& shall be in continued proportion ; and^ the JBuxion

^ of the square of ^R being equal to twice the fluxion bftne

.. area aQNo^ the fluxion of QR shall be to the fluxion of aQ as

. QN is to QR^ or as QR i^ tp Ei ; that is^ the fluxion of QR

. vhich represents the velocity at E is to the fluxion of aQ^' or

' Sa— SE, a« QR is to Eft half of the cliord EB. '

Z 4 436. Let

$60 '^ O/centripeUt Feirees. Book X.

436. Iiet the ordiaate AD represent the centripetal fbrce at
ihe distance SA^ and the velocity in a circle described about Uie
centre S at the distance SA with the centripetal focce AD shall
be to the velocity acquired by falling from a to Q in the sab«
duplicate ratio of the triangle SAD to the area aQNd, by art*
433 and 434. - And when a body is projected upwards from Ain
the right line Aa with a velocity equal to that in the circle^ it
will jufit rise to the point a, and return again from thence, if ti^e
area ADa4 be equal to the triangle SAD. When the centripe«»
tal force is reciprocally as any power of the distance whose ex-

' ponent is any number m greater than unit, there Is a limit which
always exceeds the area ADda how great soever the height Aa
may be, by art. £93 and 325, and this limit is to the rectang^
SAD (or twice the triangle SAD) as unit ia to m^—l. Hence,

. if a body be projected upwards from A in the right line Aa wiA
a velocity that is to the velocity in a circle described at^ the dis-
tance SA with the centripetal force A D in the subduplicate ra*

• tio of 9 to nt^l, or in any greater ratioj the body will rise
for ever in the right line Aa, and never return again to A. If
the velocity of a body that moves in the right line Sa, or in any

• curve AE, be at any point A to the velocity in a circle at the
^ distance SA in the subduplicate raUo of 9 to m — 1, its vdo*
' city will be always to the velocity in a circle at the same dis-
tance from S the centre qf the forces in the same ratio, because
the ratio of the triangle SAD to the limit of the area AVda
continued upwards from A is always the same wherever the
point A be taken in the right line Sa. This velocity that is ne-
cessary to carry off a body for ever in the right Kne Aa is great-
er or less than the velocity in a circle at the same distance ac-
cording as 9 is greater or less than m — 1, that is, according
^B m is less or greater than 3 ; and these velocities are equal
when m is equal to 3. When a body is projected with this ve-
locity in any other direction than that of its gravity, Ei is to
SB as 2 is to m — 1, by art 435 ; when m is less than 3, the

. body rises fpr ever in one of those parabolic figures that were
constructed in art. 392 (Jig. 171) ; when m is greater than 3,
it approaches to the centre till it fall into it in one of the
figure that were constructed in art. 393 (fig. 172^. K^rst, let

m be

Chap/XIt Of centripetal Fortet^ 861

mbe less than 3, let L( (fig. 17 1) be any point in the trajeetorfj,
*I«Pthetangent> SPperpendicuIaf toLP inP^ and letnbe to noit
as2 is to the excess of 3 above m ; let the angle LSA be made
on the same side of I^ with ISP, and in the same ratio to LSP
as n is to unit; let SA be to SL as die power of SP whose ex-
ponent is n is to the same power of SL : then shall A be tlie
jtp»s of the figure^ or the point where the tangentbecomes per-
pendicular to the ray drawn from the centre of the forces to the
' point of contact. Let AM be a right line perpendicidar to SAj
and the trajectory AL may be constructed by this right line
AM^ as in art. 392^ by drawing SM from S to any poiat in this
right line, making always th^ angle ASL to ASM as » is to
unity and S Ji to S A as the power of SM whose exponent is n it
^o the same power of SA. The demonstration is no more than
the converse of what wa^ shown in that article. For^supposiiq^
. L to be any point in th^ trajectory, A the apsis where SL and
SP coincide; and> since the velocity at L is always to the ve-
locity in a circle described at the same distance by the samecen*
tripetal foroe ,ia the subduplicate ratio of % to mr—l, by what
\x9A been shown, it follows (by art. 424)> that the fluxion of the
angle ASL is to the fluxion of the angle ASP, and the angle
ASL to ASP (art. 33) as S is to m — 1 ; so that ASL must
be to PSL as £ is to 3 — m, or » to unit; conseijuently, if
yre suppose AM to be perpendicular to SA, and the angle ASM
to be to ASL as 3~^ is to 2, the angles ASM, PSL, shaS
be equal, and SP to SA as SL i^ to SM. Let the archesyt^
kV described from the centre S through L and P meet SA in/*
and k, and the fluxion of SL shall be to the fluxion of SPu
the fluxion of the arch fL is to the fluxion of AP (by art. 21 1
and£0£), that is in the ratio compounded of that of SL to SP
and that of 2 to m — 1 : from which it follows (by the con-
Terse of art. l&J), that SP is to SA (or SL to SM) as the
power of SL whose exponent is one half of m — I is to the
same power of SA, or that SL is to SA as the power of SM
whose exponent is » is to the same power of SA; and that the
trajectory is constructed in the manner we have described from
art. 392. Because the velocity of projection at A is greater
ths^ the velqcity by which a circle is described at the same dis*

tance

«Q^le, Abe 0c«le AftL i$douWc Qf 4S»f,S4,.SM wd $L W5
iii'CimiitiHeilf fOpjartiQli^Mcl Al^^ (a^<x>piiQoa^aEal>olM^b^t^u^
An foeiVft in S >the centre ^ ^le ibrc^ ?B(ec^se 4SM ,oe7«^
.MKHiAls tore sigbt^ag^,4^(i is to.^^^btoae in^i^o tiu^tis
4Bim^ft le^ft tbui^ that «f 2 •to :3^<^j qrtbat of n to^Qxut ; ,bat
.if ^ifie;9h0NdidTsqpp9fte(a8 i»iw^ tbe^odj to go offto u^&nit^
•in tine UiilQCtoiQr (AX^ the.aqg)e 4S^ ^cnbed.by.lhe i]^y S|Ij
i^hfiUrlie ix>A:rigbt..|iDg|ei|i M^a;atip;.<^r^if tl^.q^oq^s.^S

jbl number of revolutions denoted Iff is. Tbnsj if the centripe-
tal force be reciprocally .as the power of the (^stance whose ex-

fpon«nt is Sx^, and the velocity of the projection at A be to
the velocity in ^ clrde described «t the stoie distance vrith fbe
same centripetal force in the siibduplic^te ratio of 200 to ]99»
the body will go off in 50 revolutions. If m be apposed
successively, e^qual to I J, .2 and f|, and. the vdocity of' tlie
projection be to the velocity in.a circle at the same.dUtance in
the subduplioate ratio of the number 4 to. 1^ 2.or;5> the body

. will, go off in J, ^, or one revolution, respectivfely. The bio-

^y go^ ^ff ii> A^ hypi^bolic curve when the ratio of those ve-

Ilocities i^ greater than the^subduplicate ratio of 2 to m — 1 ;
and when the ratio of tl)ose velocities .is less, the body revolves

-> continually about & tvcithin certain liinits.'

.437. When m is greater than. 3^ and the velocity of prpjec*
tion at L {Jg. 1/2) is to .the velocity.in a circle described by the

, same oentripetaiforc^ at the same distanceihtbc^subduplicate ra-
tio of 2 tOs m — 1, let. « be to unit as, 2 is to the excess of wi

. above S> make the angle I^A in the. same ratio to LSF as n is to
unit^ but on the opposite side of SL| let SA b^.to SL as the

' power of SL whx>se exponent is n is to the ^ame power of SP ;

, and A shall be the apsis of the trajectory. Upon the diameter
S A describe a semicircle SMA, and by if Ibe curve AL is to be

. constructed .(as in art, 393) by drawing any right line as SM
to the semicircle^ making edways the angle ASL to ASM as n

t!hap. XI. Cf eefttrtpekd Fonrar/ Mjg

Js to unll^ tihd'SL to SAasthe.|mferof SM "ivboie ^sipoMat
is n is to the same power of S A : the demon^fratioB is simile
to that in the last article. When theeentlripetal ibl0te « leci*
procally as the 'fourth power of dte'dbtanoe, ^ ib^qnal to 4;
•and if die velocity of the^rojecfion be to the velocity in a<3kole
•at the sailie distancieas'the sq^orerootof £ isio the squaie loot

*bfS^ then n is equal to ^2, the angle ASL is double of 'ASM«
the right lines SA, SM^ andSL ite iii-continttedlproportionyaad
the trajectory AL is an epicycloid that isdesotibed by afpoait
in the citt^umferenceof a circle while it lerolfesnpdn an eqofi
circle^ as We^hate showh ^bewhdfe. When «s is equal to S,
1& is to ITS — 1 as 1 to je;'andif the Velocity of tbefNr^geciioa
be to the T^iotify In tike cirde-ss 1 lo tte jsqueie mot^f 9j n
iseqiiaHounit/ahdUie'«emlcirtleHs6ifisibetm)6ctoiy. When
m is equal to7^ fi isto^^»-*-ra8'l to 5; add if the veloci^
of the 'projection be to the velocity in the circle as 1 to the
squioe root of 3, ndiall Be'eqoaltpi^ Or ASL shtdlbeose
tialf of ASM^ SLdiaU be a ineaji))ffdp^onia betwixt SA and

* SM^ and the ttBJecfibfy is the tame I^Ui that is called the
lemniscaia by the eetebrftted Mr. £t$fkouUi. It is manifest

^that &e body falls ni^mys into the centver S in ^ aaaiber of

' inevolntions from thei^tpbis A dtaotedby.^^^or by l^if the

fraction - be equal lo* the eirc&s'ifw ftbirve^S. 'Thiis, if bs

' 'be iqasl to 3 t^/aad the Velocity of projection at'A be to the

trelocity in th^ citde at A in the subduplicate ratio of 200 to

^l^ the body win fall into^the centre after 50 revolutions^ and

the number of such revolutions is inversely as the excess of m

above 3. The sanie constructions of the trajectories 4n this

and the preceding article were^ given long' jigOj Descripjt.

eurvipar. 2, ptop. %2.

438, When w is eqtal to^S ifig. IQlOi ^^^ ^^ velocity of the

"* projection is ^cfual to the Velocity by which a circle is described

- at thesame^isiltncewithtbesakn^t^lfntripetalforce, theaogular

' Velocity of SE'i^ equal to the an^lar velocity of SP ; thcre-

. fore the angles ESP, SPE are invariable, and the trajectory is

the loganibmic spiral in which the body will approach to the

centre S^ oriecedefrom it> aoc'ordipgas the direction of the

pro-

1

:S64 Cf uktrtpttal Fortia. : Book t

projection fomis an acute or ra obUue angle with the raj drawn
to the centre S«

439* Whem m b eqnal to nni t^ or less^ there is no assignable
space which the area KDda may not exceed by producing Aa
upwards (by art. d£5)> and a body projected upwards from A will
return again how great soever the velocity of the projection
may be ifig. 192). If we suppose a centriiugal force to be air-
ways directed from the point S that is inversely as the power of
the distance whose exponent is any number m less than unit^
and a body to be projected at any pwit L with the velocity
which it would acquire by this centrifugal force in moving fiom
S to L, or Ihatis to thevelodty in a circle described at thesame
distance by a centripetal force equal to this centrifugal force
in the inbdnplicate ratio of fi to 1 — m, the trajectory will be
constructed by the right line AM io the same manner as in art.
4S6^ only, in this case^ n is a fraction that is to unit as £ is to
3""Hn. Thus, if the centrifugal force be invariable, and the
ratio of these velocities be that of the square root of £ to unit,
the trajectory will be constructed by taking the angle ASL
always equal to two thirds of the angle ASM and SL equal to
the first of two mean proportionals betwixt SM and SA $ for in
this case » is to unit as £ is to 3. If the centrifugal force be
as the distance, n is equal to |, ASL is one half of ASM, SL
is a mean proportional betwixt SM and SA, and the trajectoiy
is an equilateral hyperbola that has its centre in S. In all
those cases the trajectory hasan asymptote that passes through
S, and constitutes with SA an angle which is to a right one as
£ is to the excess of 3 above f».

440. It may be of use, to avoid mistakes that may arise in
enquiries of this kind, and in other cases where second fluxioni
are introduced, to reflect here on what was shown concerning
these fluxions above, in art.74,97» and in several other places.
The centripetal fbrcein the curve AEM (^g. 191) at £ is most
commonly measured by TM the subtense of the aagle of con«
tact at E when the tangent ET is supposed to be diminished
infinitely, the time in which the arch EM is described being
given ; that is, by Tr when ET which represents the velocity
at £ is of a finite magnittsde (a? we always suppose it)^ apd

Er

Chap. XI. Of amtripeial RtUb. 5S5

£r isa parabola of ihesaiiieeimratare at Evith ABM r because
Tr is the space which the body woald describe by ^e gravity .
at E continued uniformly^ in thesametime thatthebody would
describe ET by its motion at E continaed uuiformly in the
tangent. If therefore we suppose QN (which represents the
centripetal force at the distance SQ^ or S£) to be .equal to Tr,
thesqaare of the velocity atEwillberepresented by therectangle
contained by EB (or £E&) and QN> becaufle the square of ET.
is equal to the rectang^ cocilained by EB and Tr. And, since,
the square of the velocity that is acquired by falling through.
£i with the same gravity QN continued uniformly, isrepre^
sented by twice the rectangle contained by E& and QN (art.
434), it should follow, that the velocity in the cnrve at. E is.
equal to the velocity that would be acquired in falling through
EA by the gravity at E continued untfonnly ; whereas it has,
been shown (art 433), that these velocities are not equal, but
are to each other in thesubduplicate ratio of l to2. In the same
manner, the velocity in a circle at the distance SE would be
found equal to the velocity that is acquired by faUing through,
the radiusESwith the gravityatEcontinned uniformly ;wherea8
we found this velocity to be greater than the velocity in the
circle in the subduplicate ratio of 2 to 1 . The error arises from
the supposing thegravity, when the body descends in the right
line ES, or aQ, to be measured by the fluxion of the velocity
of the body, or the second fluxion of the space described by it^
and at the same time by therightlineTr ; forthesesupposit&anf
are inconsistent, because it is £Tr, and not Tr, that measures
the second fluxion of the space described by the body while it
descends in the right line aQ with a motion uniformly accele*
rated, as has been shown by several di£forent methods in art.74>
75, 97> & 254. The gravity is measured by the increment of
the velocity that is generated in a ^ven time when the gravity
is invariable (and, consequently, the velocity increases unt^
formly),orby the difierence of the spaces that would bedescribed
in the given time by the velocities at the beginning and end of
that time continued uniformly: but Tr is equal to the half of
this diflerenceonly, being equal to the excess of the space that
is described ijx the same given time when the motion is uni*

formly

fmnly acoabrticd aboiNi Arqp«M lfc%l wauU bi^?» b«^ de-
scrib«d if the motioa ka4 ocviitipMd MiiiMm item the b^iii«
ningof thattiiae. ThefflAnof^ in ai^dcr to A?pid 99ppo«tioii»
that are ipconmtsnt^QN vAiiqfc inisaiaji^th^i^tripftfl f<||ce
at Q, or B, filial be a9pp9iQd epul «a STr^ tb^ v^ocllj at £
ift theovnie it to be ittpref^fitafl bj a mean firoQartilpoal be-
twixt 4QN aa^ £B» fvQ^ii aai £6, ap4 the ^^IiipU; ia a ctr^
ele at the ^ifUnoft fi£.hgF A^n^ao piaportioiiiil betwirt QN 4]»4
the raditts-fiE. Dm ezaj^h vro hwe drnvib^ IQ119 s^eirve to
show hovr mifttaloBi of ^ia kbd have sometiiBies at^o in ih^
i^^catioB of this methfid^ and kov tbey m^ he i^voided*

441. WhenaisirokJ8dittcrihedbyftC!^ntripe<iU|9rcedire^
ed towaidsits centre^ t^enottoo in the pircle^ the angubir mo*
tiop of the my drawn from, the eeatre tp tbi^ revplvipg ho^y,
Ijhe motion with .w^ich the acea £aw8 th^t k d^^crtbed bjr thi|
ray^ and the angular motion of the taught »t the bady>»ie all
mitform. In ottier rases, the aie& de^crib^d by the ray S£
drawn fron the centre of the forces S to the sevolffii^ body
at'dl flows uni&inaly^ and measmes the time ; but the v^lociif
at any poiiit E is iiwevsely as SP the perpendiQuter 6sm S oa
ihe tangent ET, the aagukr Tolocityjof the rvy ££ eboHJt S ia
ittTersely as the square lof SE^andihe aogular velocity of tht
tangent (or of the ^M^tfieadiodacSP about S) is inver^y n^
the reistang^le contained by SP and .the ray of curvAtqre at £.
Sir JsMe NMtfM'a-donoBStraAiQn of Abb lofigr be refimenbed in
the foUojviring nunner. \When» body moves in njrigbt line £T

(fig. 1^5) withnnaaifonuau»^n#itdescribeseqwl«p«oe&onthAt
line in aoy eqaal twines ; andif a x^be supfKwed to be drawn
silways &oin4be body toa^pben point S tb»t ia not in thai
right line, this ray. shall dncfBie cgual aima about S in any'
equal times, Elem. 35. 1. Jk fonce. tt^.acts upon the b9dy aik
anj point Ediieeted towardsS haa.no ^fl^ton die m^^tiide
of the wea des^pdbed in a given tifie by^the right Jine draw*
always from thebody to.S : it may^aooelerateor lefard the
motion of the body; iMittfaeareadeaQribedabont^ls of the
same magnitadefur if no audi fpnce had acted npon it. Por^
iet ^ErT be to-BK as the .vdodty .pf the body in . the.dinection
1BT i8;to*«he ^ebbity^thatjrbuld bp pmduood by the new imr

pulse

pulse in ttie dlfection £S, cdmpfete ft^ fyftraHelogram CKMT^
join ?r and SM ; tbe body tcill now describe the diagMal EM
in tlxe same tim^ that it wdtild have tiescHbed the side BT if
jnio new force had acted npoti it at B^ find ttie tiungle S£M in
equal to SET. The ^sme h to be said of waj {nitoeBdite im-'
pulses provided they be dlways directed to the swftie ceiitm S4
they never affect th<& flu^^ion of ibe area> #hicih thetitfore flow*
in the, same maniie^ as If the liody had j^toeeeded in th« tan*
gent with £he motion at £ (Ji^. 191) continued Mtfonttfy* It
follows from this, that if AL be th^ tangent at A^ atid the tti*
angle SAL be eqtk&l to &e ttian^le S^^ &e» sAL oxid tX
would be described in 6qil^' times by the motions td A and fi
continued uniformly. Thei'efet^^ if SX be pevt^endiMlar to
AL in X, the velocity at A Shall be to the Telodty at £ as AL
is to E?r, or as SP is to SX ; 'that is, therdbcity ih any giveti
figure is always ih^r^Iy as the p^rpeiidicidiu: frdtt thetoatie
of the forces on the tangei^t.

44e. This maybe likl^wi^ demobstMed from att. 496, wheft
ilhvas shown, that if the Square of QR foe equal, to twice tha
area aQSd, so Ihat Qft may i^preseiitlhe veIoc% at B, tba
duxion of Qtlbhallbe to the flnxion of aQ as ^ Is to £i«
The fluxion of SP is to the fluxion of S£, or SQ, as SP k td
Bb, by art. 334; aiid the velocity witli which SQ'dedre«M is
aqual to the velocity wHih "Which oQ in6i«aMs ; therefom^ the
velocity with whfc'h 'SP d^ti^ea^s is to that with Which QR 4&*
creases as 1^ is to QlR ; and the fluxion of the reetanf le eoQ«
tained by tStR &nd SP vanishes, by ^rop. S ; consequently, this
rectangle is invariable, and QR ttrfa¥cfa measures 1^ yelocityat
E is inversely asthe'peipiBridlctilarl^. Because theareadescrib*
ed aboutS flows uniformly mid measunes'the time, tbeangu^
lar velocity of SB about S is itciptocally as the square of the
^stance SE, by art. 120 (Jig. 32, n. !2); imd'if the siector ESI
be equaT to the ared £SH, the i^ctot ESI will be generated by
SE, if its apgulaf motion about S be continued tinifennly ia
the same time that the area ESH is described by the ray drawn
Irom' S to the body while it mov^s in the arch EH 0ig. I9I).
The-angular velocity of the tangient at £, or of the perpendi-
cular BP, is to the angular vdocity of SB as ISB* is to Bk,

aOi O/eenttg^dlForeeL . Bookf»

•nd is inveraely 9B the rectangle SEA, or the rectangle con«*
tained by SP and the ray of cunrataie EC.

443. The centripetal force at any point £l is as the sqnaf d
•f the velocity at E directly^ and £& half the chord of thei
circle of curvature that passes thrdugh S inversely, by art. 422.
Tlierefore this force is inversely as the solid contained by £6
and the square of the perpendicular SP, or (because E6 is to
SP as the fluxion of the distance SE to the fluxion of the
perpendicular SP> by art. 384) inversely as a solid that is to
the cube of SP as the fluxion of SB is to the fluxion of SP. It
is likewise as the angular velocity of SP (or of the tangent at E)
directly, and a third proportional to SE and SP inversely. The
same force may be measured by the diflerence of the second
fluxion of SE considei^d as terminated at the tangent Et, and
its second fluxion considered as terminated nt the curve EH,
or by the sum or diflerence of the centrifugal force that arises
from the motion of rotation (which is always inversely as the
cube of the distance SE, by art. 442, since the angular veldcity
of SE is reciprocally as the square of SE) and the force by:
which the motion of L (fig. l66) is accelerated dr retarded at
E in the right line ES, as in art. 43 1> and by several other
theorems of this kind.

444. When a figure (Jig. 19 1) has the same curvature and
the tangents inclined in the siame angle to the rays drawn to
the centre of the forces at any two points, the centripetal
forces at these points are reciprocally as the squares of the
distances. For, since the chord of the circle of curvature is
thesame, and SP is to SE in the same ratio in both cases, the
solid contained by EB and the square of SP is as the square
of SE. Thus, the centripetal forces are reciprocally as the
squares of the distances at the extremities of die same axis in
any conio section, when the centre of the forces is any where
in that axis, or in a circle at the extremities of the diameter
that passes through the centre of the forces.
445. Let AEM (fig^ 194) be an ellipse, and the centripetal force
be directed to the centre of the figure; then, since EB b equal
to the parameter of the diameter that passes through E (by
art. 374), the rectangle contained by the semidiameter OE and
£6 is equal to the square of the semiconjugate OK. The rect-
angle

Chap. XL Of eenhipetal threes^ ^f

«ngle contained by OK and OP the perpendfcnlar from O on
the tangent is invariablef. Therefore OE is inversely as the so-
lid contained by the sqnare of OP and E6 ; and the centripetal
force is as the distance OE. This is likewise easily deduced
from what was shown in the introdtiction, p. 8. that^ when
the area EOM described about O (and^ consequently^ ihe tune
in which the arch EM is described) is given^ then TM is in an
invariable ratio to the distance OE. The velocity at E is as the
semidiameter OK conjugate to OE ; because the petpendicular
from O on the tangent at E is inversely as OK. The periodic
time in the ellipse is equal to the periodic time in a circle de-
scribed at any distance OE by the centripetal force at E : fbr^
if the tangent Et be supposed equal to the circumference of a
circle whose radius is OK^ the triangle "EOt shall be equal to
the area of the ellipse^ and the periodic time in the ellipse equal
to the time in which E^ would be described by the motion at
£ continued uniformly, by art* 441; bnt the velocity at E is td
the velocity in the circle at the same distance as OK is to OE^
(by art. 425)^ or as E^ is to the circumference of the circle ;
consequently^ the time in IBi, or the periodic time ip the ellipse^
is equal to the periodic time in the circle. It follows^ that the
periodic times in all ellipses are equal (as well as in cirdes^
art. 432)^ when the centripetal forces are as the distances. Let
OB be equal to OQ^ and the velocity at any point E equal to
the velocity that would be acquired by falling from ^t to Q ;
then shall On be equal to AH the distance betmxt the extremi-
ties of the transverse and shorter axis. For, if ad and QN re-
present the centripetal forces at a and Q, as in art. 435, the
trapezium aQN(2 shall be to the triangle OQN in the dupli-
cate ratio of the velocity at E in the ellipse to the velocity in a
circle at the same distance, or as the square of OK to the square
of OE ; and the square of Oa to the square of OE as the sum
of the squares of OK and OE to the square of OE ; conse-
quently, the square of Oa is equal to the sum of the squares
of OK and OEl^ or to the square of AH. Hence;.if a body
be projected at A in a direction perpendicular to OA with the
velocity that is acquired by falling from a to A, and a circle
described from the centre A with a radius equal to Oa meet
VOL I. A a OH

1

ank Of centripetal ForeeK Book t.v

OH perpendicular to OA in H> then OA and OH shall be
the two semiaxis of the trajectory^ In the same manner it is
8hQwn> that when a centrifugal force directed from O is as the
distance, an hyperbolais described that has its centre in O: and
as the time of the motion is measured by an elliptical or circa-
lar.area when tiie centripetal force is as the distance, so it is
measured hgr an hyperbolic area when the centrifugal force is
as the distance ; that is, by the measures of angles in the for-
mer case, and by the measures of ratios in the latter ; which
is agreeable to what was shown above in art. 159 and 407 •

446. Let the centreof the forcesbe now in the focus S C^. 159) of
any conic section ; let the circle of curvature at E meet £S in .
B, ^ and BX parallel to the- tangent at £ meet EX perpendi-
cular to the same.tangent in X ; let XZ be perpendicular to
£B in Z, and £Z shall be .equal to the parameter of the
transverse axis of the figure, by the 6th property of the circle .
of.curvature, mart. 375. Because the triangles SEP, BEX
axe similar^ the square of SP is to the square of SE as the square
of EX to the square of EB, or as £Z is to £B ; consequently,
the solid contained by EB and the square of SP is equal
to .the solid contained by EZ and the square of SE. There-
fore, since £Z is invariable in a given figure, the centripetal
force towards S is reciprocally as the square of the distance SE.
It follows from what was shown above (art. 434 and 4fi7)> that
the velocity in the parabola is to the velocity in a circle at the
same distance as the square root of 2 to unit, that the velocity
in the ellipse is to the velocity in such a circle in a less ratio, and
that the velocity in the hyperbola is to that velocity in a great-
erratio(^g. 195). Tlietimeinwhicharevolutioniscompletedin
an ellipse, is equal to the time in which a circle is described by ,
the same centripetal force at a distance equal to half the trans-
verse axis ; for the velocity at H the extremity of the shorter ,
axis is equal to the velocity in this circle, by art. 426. But if ^
HT be supposed to be taken upon the tangent at H equal to the
c^ircumference of that circle, the triangle HST shall be equal to
the area of the ellipse, and the periodic time in the ellipse equal
to. the time in which jtIT would be described by the motion at
H contiAjoed uaiformly in the tangent (by art. 441)> and, con- ^

sequcntly.

^hap. XL Of centripetal Forces. S7i

ieqinstetly^ equal to the periodic time in th<s.^oirdle described at
th« distance SH. Hence^ when ellipses are described by cen-
tripetal forces that are always inversely as the squares of thedis*
tances from their common focus, the squares of the periodic
times are as the cubes of the transverse axes orof the mean dis*
tances. Andsincethetransverseaxisisdetermined when the ve-
locttyand centripetal force atanygiven point in the curve are
given (by arti 4fi7)» it follows, that all bodies projected from a
given point in different directions, but with .equal velocities^
complete their revolutions, and return to the. same point again
in equal times. The transverse axis was determined in art. 427.
The shorter axes of such ellipses are to each other as the per-
pendiculars from the centre of the forces on the right lines in
which the bodies are projected, or as the sines of the angles
contained by these right Imes and the ray drawn to the centre
of the forces ; because the square of SP (fig. 159) u to the
square of SE as EZ is to EB, and (since the transverse axis with
the diMdnce SE ahd chord of curvature EB are given) the
square of SP is as the parameter EZ, or as the square of the
shorter axis (fig. 195). Let SE be equal to SQ^and the velo«
city in the ellipse at E be equal to the velocity that would be
acquired by falling firom a to Q, and Sa shall be equal to the
transverse axis of the figure. For, let SA be taken upon Sa
equal to SH,and A/ be the ordinate of the figure ai/NQat A;
then, because the velocity at H in the ellipse is equal to the
velocity in a circle at the same distance, the triangle SA/ shall
be equal to the area ahld (art. 436), or to the difference of the
rectangles Shi and Sad ; consequently, the triangle SA/ ia
equal to the rectangle Sad, and ItSd is to SA as A/ is to ad, that
16, as the square of Sa is to the square of SA ; and Sa is equal
to SSA or to the transvme axis of the figure. Therefore, if a
body was projected directly upwards from E in the right line
SE with its motion at E, it would rise to a distance from S

•

equal to the transverse. axis of the figure;. and the motion
with which acircle is described would carry the body to a dis«
tance from S equal to the diameter of the circle. If a body
begin to fall from a in the right Une aS, it will be carried to an
equal distance f^ntS on the other side, and will thence return
• .... A a S: again

i7ft 9f ctntiipetol B9foaM BookTi.

^ain I6« in the same time Uiat arevoluticft W oomi^letecl ia
tii€ eHip8e AEa whose tranvveive axb is equal feo Snj and the
time in which it fiiilis from a to S is one feiftth pavt of this
Hme, or of the ftcriodic time in the d^se AaA# or of the
tune in which a cifcle ia described at a distanee finom S eq^gl
to one half of Scr. If a body be prelected from A in a right liiie
perpendicnkr to SA with the vdocitj that would be acquamd
by fidting ifom o to A> then A shall be an. apsis of the fignn^
and Sa the distance of the other apsis shall be equal to oAu
When the velocity at £ is to the velocity in a circle at the
same distance as S£ is to SP^ the angular velomty of thengr
SE is equal to the angular velocity of the ray in a oiitde de-
scribed by the same centripetal force at the same distance : ia
this case SE is equal to one half of the prinaqpal parameter of
the figure, the transverse axis isperpendicular to SE^ the tnyec*
lory is an ellipse;^ parabola, or hyperbola^ acoording as the
angle SSP is greater> equal to, or kss than half a ri^t one %
and the paracentxic velocity in the trigectory is alwap greatest
at this point E. Because the primary planets move in ellipses
that have their focus in the dentre of the sun, so that the arean
described by the ray drawn from any one planet to the sun in
equal times ave equal, and the squaies of the p^odic times of
the different planets are as the cubes of their mean distances
firom the sun (as was observed by the famous Kepkr\ and be«
cause the same laws dtitain in the motions of the satellites
about the primary planets. Sir Isaac Newttm concludes^ that
there is a centripetal force extended over the system to all
distances directed towards the sun: that decreases in propoc-
lion as the square of the distance fWnn its centre increases,
and that a force decreasing in the same manner is directed
towards each body in the solas system which at equal difr-
tances from their centres ia as the matter in each of them.
By comparing the ^fects of these forces with, the descent of
^avy bodies near the surfsce of the earth, he finds, that the
power of gravity so well known to us is one instance of this
general prindfde. The powers upon which the celestial m<^
tious depend being thus discovered, and reduced to knowa
measiMtts, he then pioceads to deduce these motiosa with their

various

.C%«|K^L Of€6Rlrlpatd Form. S7S

taiious iiKsqaaBtifeB fitHa tbek pnndplea; ancl a Mer ac-
conat of th» matter is expected fiom an aa^or who has
already ghrea aofeial proofs of the great ptogress be has maafe
ia thia most uaeliil theory.

447. Some find it dtf&cnh to concern hovr a body can le-
v<rf ve in an eUipse^ and, after approaching towards tike centre Of
the forces in deacendiag firom the higher apsb A to the lower
apeis a^ recede from it again by ascendii^ from the lower apsis
a to the higher h, since the centripetal force is greater at a than
at A. There are indeed varioas laws of the centripetal fbreetf^
which would either canae the body to desoend frota the apsis
continually towards the centre^ and at length fall into it, or to
;ucend from the apsis cDnthpaaUy and recede from the eeatre
for erer : but there are other lai^s which if the oeatripetal force
obserye the body may appiroadh to the centre, aild reeede from
it by turns \ and of this kiad is the law which obtains in the
^\fa^ system. In oxder to di^nguish those cases fiotti Meh
other, we are to obsenre, that when the vdocity oS the boify
at the high apsis .A ia less than that iThich is requisite
to carry it in a circle about the centre at S at the saihe distance
SA> the body mast move in a carve tliat frdb within thateiick
theie^ and most approach tov^ards the centre; whOe tt descends,
its vdocity increases ; and if its veiocitj increase ill a higher
proportion than that in which t^ velocities re<{aisite to Oan^
bodies in circles about S increase, the velocity in the low^ part
of the curve may at length exceed the veloeity ia a circle k\ the
same distance, and thereby beocmie sufficiexvt to carry oiFthe
body again. In these cases the velocity of the body m^eorbt^
and the vdocity in a circle described at the same distamce, may
€:tceed each other thns by tdra;!, the latter ia the higher
part and the former in the lowevpartof the orlMt, and thel^dy
may approach towards the centre, afad reeede fron it by tm-nii.
Thus, in ^e solar system, the centripetal force ineieases as the
s<)aiure of the distaneefrom the cenilre decreases, andtbe velocity
tbat is requisite to carry a body in a circle about the centre in^
creases only in the sabdupUcate ratio in which 1^ distance de-
ciMses, art. 43i^ but the velocity intbeovbitincTeasesinahigher
proportion while the distanoe decreases (by art. 441), so that
Ihoa^ tbe former exceeds the latter at the higher apsis A, the

A a 3 latter

'S74 O/centr^etalJI^rca. Book L

latter by increasiiig in a higher proportionbecomes equal to it
at the mean'distance SH, and exoedls it in'4tal;um at the lower
apsis a ; and thus the body constantly revolves from one apsis
to the other. This is further illustrated^ by compann^the cen-
tripetal force in the orbit with the centrifiigal force that arises
from the circular motion of the body about S ; for this cen*
'trifugal force increases always in proportion as the cube of the
«[istance from the centre decreases^ and consequently in a.
higher proportion than the centripetal force ; so that each of
.these fbrces prevaUs in its turn, the centripetal force in the
higher part, and the eentrif^ugal force in the lower part of th^
orbit,

• 448. In general, a body may af^roach towards the centre S,
and recede fiom it by turns, when the velocities that are requi*
* eite to carry bodies in circles about S increase in a leas pro-
portion than that in which the distances decrease. But if the
centripetal force be such, that these velocities increase in the
aame piroportion in which the distances decrease, a body can-
not revolve about the centre S in this manner ; if it begin to
approach towards the centre S when it proceeds.ftom the apsb
A, it will approach continually to th^ centre till it fall into it ;
or if it begin to recede from S when it sets out from the apsis
a, it must ^cede from the centre to greater ai^d greater dis«
'tances for ever. For, should we suppose the body to descend
in this case from the higher apsis A to the lower apsis a,
since the velocity in the orbit at A would be to the velocity at
a as Sa is to SA (by art. 441), that is (by the supposition),
as the velocity in the circle at the distaace SA is to the tcIo*
city in the circle at the distance Sa, it follows, that the velo-
city in the orbit at A would be to the velocity in a circle at the
same distance as the velocity in the prbit at a is to the velocity
in a circle at the distance Sa; and that, since the velocity in
the orbit at A is less than the velocity in a circle at the dis-
tance SA, the velocity in the orbit at a would likewise be less
than the velocity in a circle at the' distance Sa ; so that the
body would continue to approach to the centre after it comes
to a. In the same manner, if the velocity at a in the orbit
exceed thp Telocity in a circle at the distance Sa, the velodty
in the orbi( at A would likewise exceed the velocity in a

circle

.Chap. XI. . Cf centripetal Ibroen. 875

circle at the distance SA^ and the body would oo&tmne
to recede from S after it comes to A. When the oen-
tripetal force is rec^rocaUy as the cube of the distanca
ircnn S^ the velocities that are requisite to carry bodies
in circles about S increase in the same proportion that the dia^
tances from S decrease^ by art. 4S£ ; therefore the body after
if proceeds from the apsis (unless it move in a circle)^ must ap*
proach continually to the centre^ or recede from it for ever^ and
the figure can have no more than one apsis. When the centri*
petal force is reciprocally as any higher power of the distance^
the velocities that are requisite to carry bodies in circles about
Sincreasein ahigher proportion than thatin which thedistanoes
decrease ; consequenilyj the body after it sets out from the apsis
either approaches continually to the centre till it faD into it^ or
recedes from it for ever. Thus it was shown^ in art. 440> that^
when the centripetal force is reciprocally as a power of the di$»
tance whose exponent is any number that exceeds 3 by any

fraction -, if it be projected at A with a velocity that b to

the velocity in a circle at the distance SA in the subduplicate
ratio of ^ to 25-|-l^ it will fall into the centre in a number
of revolutions denoted by ^. But when the centripetal force
is reciprocally as a power of the distance whose exponent is less
than 3y the velocity in the trajectory increases while the distance
decreases in a higher proportion than the velocity in circles de*
scribed by the same centripetal forces^ by art. 441 ; and in those
cases the body may approach to the centre and recede from it
by turns.

449. In general (Jig. 191), the velocity in any orbit becomes
equal to the velocity in a circle described at the same distance by
thesame centripetal force when the angle ESPisamaximuTTi^ or
when the angle contained by the tangent and ray drawn to the
centre of the forces is apiinimumi by art,424j or supposing (as in
art. 435) that the centripetal force at any distance SQ (or SB) is
measured by the qrdinate QN^ apd that the velocity at A is
that which would be acquired by falling from a to A, when,
the area cQNd becomes equal to the tfiangle SQN; that i^
if the centripetal fprce be reciprocally as the power of the di^-
4ance whose expbnentis m, wheh the rectangleSON is to the
rectangle Sad as S is to 3 — m, or when the power of Sa whose

A a 4 expo-

S7S , €>fcenirip€talF^cei. Spolrl.

exfcneotU M'^i is to the same power of SB in lliat mtiou
Let a be the other apsis of the tcigectory^ and^ S6 bdng tnada
equal to Sa^JetOF lepiesentthe oentripetal foioeat the diataa<:e
SO (or Sa)^ and SA shall be to S6^ in general, in the sidMhi*
]ilicste ratio of the aiea oCrFJ to the area oADd (bj art, 4SS
and 44s)) that is, in the present supposition, in the sabdupliDale
ntio of the diference betwixt the rectangles SGF and Soul
to the difference of the rectangles SAQ and Sad. The ango«
lar .T^ocity of the ray SE in the trajectoiy becomes equal to
Ihe angular velooi^ in a circle described by the same centripe*
tal fonae at the same distance, and the paracentric velocity ia
l^eatsst, in general, when the velocity in the curve is to the ve«
lodty in a circle at the same distance asSE is toSP, or when
fiEis toSA in the sobduplicate ratio of the area aADd to the
Jsiaogte SQN (jig. 196). Thus, forexample,if thecentripetal
fiwoefae the sam^e ia all distances, the velocity in the tn^toij
^eoomese^al to the velocity in a circle when SE is two thirds
pf Saf md Sa (or SG) is to^SA in the subdupUcate ratio of oG
to a A, from which it follows, that, if a A be bisected in K, and
a circle described through K from the cei^tre S meet AL per*
pendicular tp SA in L, then Sa shaU be equal tp the sum ofKA
anfl AL^'and A is the higher or lower apsis according as SA is
greaier or les$ iban this sum ; but when SA is equal to this sum,
a circle is described about the centre S. When the cube of SE|
Is dbubl^ of the solid contained by aA and the square of SA^
the angular velpcity of the ray S£ is equal to the angular veloci-
ty bf a ray in a oircle that is described at the same distance by
the same centripetal force, and the paracentric velocity is then
greatest at E.

450. In thelogarithmic spiral, SPis to SEC^g. ipOinan in-i
vi^able tatio,EfrisequaltoSE,andthe centripetal force towards
6 Is inversely as the pube of SE, by art. 443. In the reciprocal
or hyperbolic spiral (described in art. 344), ST is invariable ;
fipom which it folbws, that the fluxion of SP is to the fluxion
of $E as the cube of ST to the cube of ET, or the cube of
SP to the cube of SE, and that SP is to Eb in the same ratio ;
eonsequentiy, thecentripetal force towards S is inversely as the
cii))e of th$ ji^tance SE. The centripetal forces observe Xhf

same

tame law when ihe squaie of SE is tp the tqiwe«of Sl^-as thfs
the snm or d^erence of ai)y giTeu qpace^ and the square of S^
is to ^ given square. The constniotionof the figures wbifib hay^
this propertyisgiveD^ Harmon.mem^r.p.Sl,9f^PhilQ$. Tram^
n. 317* ifig' 17 !)• When any figure conatnicled in art. S9ft. ^
described by a centripetal force directed towards S, this force ia

inversely as the power of the distance SL whose exponent j#

s
S^^ , because LI is to SL aa fi is to i»-^l .SP to SL as fiAis-lo

SM^ and consequently^ SP always as the power of SL whose
exponent is i — -. (Jig. 172). When any of the figures con-
Jbmcted in art. 393 arc deacribod \^ a force *^vfctied to*
wards S^ SP is as the power of SL whose exponanl is ii»f>JL»

LI is to SL in an invariable ratio^ and the force is recmropalljr
as the power of SL whose exponent is 3:4- ""i 9pA <fai9 js il^^

iponverse of what was shawn above in art. 4S6. and 437«

451. Let AM (Jig. 173)'be any curve line that can be de^
scribed by a force that is as the power of the distanoe whose ex-
ponent is any number m, let the »igle ASL be to ASM as m-f- 8
ts to 2^ and SL beto SA as tfaepowerof SM^ whote expooentis
onehalf of m4-3 is to the same power of SA ; tfaentbecarve Ah
may bedescribed by a centripetal force directed towards S, t^Mft

is as the power of the distance SL whose exponent is -rx — ^*
The demonstration may be deduced from art^ 394^ but will ap*
pear more easily afterwards. ThuSi if »i be supposed succes*
sively equal to 8, 1, i, 0, — ri — J, the curvje AL may bp
described by a centripetal force directed towards § that is ift-
versely as the power of the distance whose exponent is 2|., ^
If' li> li> H' respectively. The point A is an apsis com-
mon to AM and AL. If B be the other apsis of AM and Q
the other apsis of AL, the angle ASD must be to ASB as m+S
is to 2^ by the construction : Therefore^ if M — 3 be supposed

^nal to m, a)^4 N— ? equal to;^^-^^ theiuE^te ASH

•Ml

378 Of the Resistance and Book d

shall be to ASB in the subduplicate ratio of M to N. And this
is agreeable to what is shown by Sir Isaac Newton of the
motion of the apsides (Princip. lib, I, prop. 45. ex. 2), whea
the excentricity of the orbit is supposed incomparably little^
which case only he has considered.

452. Suppose now that the revolving body describes any tra«
jectoiy AEM in a medium that resists its motion (fig. 196).
Let the centripetal force at E by which the curve would be
described in a void be to the centripetal force at E by which it is
described in the medium as any given invariable right line Sa
is to SZ^ then the resistance at E shall be inversely as a space
that b to the square of SP the perpendicular from S on' the
'tangent as the fluxion of the curve AE is to the fluxion of SZ;
and the density of the medium (supposing the resistance to be
as the density of the medium and square of the velocity toge-
ther) shall be reciprocally as aright line thai is to SZin the same
ratio. For let the velocity at £ in the void be represented by
£/ and the centripetal force by EAj the velocity at £ in the
medium by £L and the centripetal force by EK^ the resistance
by ER^ and^ the fluxion of the curve being represented by EN,
letNV be perpendicular to the taqgent in V^.tfaat £V may re-
present the fluxion of the ray S£. When tne directioa of the
motion at E is in the right line £T that forms an acute angle
with the ray S£, the velocity is accelerated by a force that is to
the centripetal force EK as EV is to EN the fluxion of the curve,
and is at the s^me time retarded by the resistance £R ; the
rectangle contained by 2EN and the difference of these two
forces (that is, the excess of the rectangle 2KEV above the
rectangle 2NER) measures the fluxion of the square of EL, by
art. 424 ; and the fluxion of the solid contained by Sa and the
square of EL is measured by the solid contained by Sa and
2KEV— 2NER. But the square of EL is to the square of £/ as
EK is to E* (by art. 429), or as SZ is to Sa ; and the solid con-
tained by Sa and the square of EL is always equal to the solid
contained by SZ and the sqiiareof £/; consequently, theirflux-
ions are equal, and the solid contained by Sa and 2KEV —
2NER is equal to the fluxion of the solid contained by SZ and
the square of EV ; but this fluxion consists of two parts, the

solid

.Cha(>;Xr« I^mty of ike Medium. 979

solid oontnined bySZ and the spacewbich measures the flaxioa
of the square of E/ (that is, the solid contained by SZ, 9JEM
and £V^ by art. 4S5,ot the solid contained by Sa,2,EKy andEV,
since SZ is to S« as EK is to Ek), and thesolid contained by th^
square of El and the fluxion of SZ. From which it appear^
that this last solid is equal to the solid contained by Sa, SNB,
iand £R; but that the fluxion of SZ is negative, or that SZ
.must decrease while the body approaches to S in the arch EM
and the ray SE decreases. Therefore, if Zz measure the
Haxion of SZ, tlie rectangle contained by 9Sa and EN shall be
,to the square of £/ as Zs is to ER that measures the resistance
* at E ; and, because El that measures, the velocity in a void at
£ is always inversely as SP, it follows, that the r^istance at E
is directly as 2a the right Une. which measures the fluxion of
SZ^ and inversely as the solid contained by the square of SP
and EN the right line that measures the fluxion.of the^curvie
AE. MHien the direction of the motion at E isin theright line
.£/ that forms an obtuse angle with the.ray SE, .the velocity is
^retarded by a force that is to the centripetal fprce ^K as EV is
to EN, and at the same time by the resistance ER ; and it will
appear iji the same manner, that the resistance ER is directly as
^> and inversely as the solid contluined by tbedqiiareof SP
jand EN, but that in this case SZmust decrease while the r^
SE increases, and the body recedes from the center. .Nor oaa a
body move in the trajectory, eithei:by proceedij^ from B t<K
wards M or towards A, unless.SZ beg^i to decrease fixNii the
moment when the body sets out from £. If the area ASP be
supposed to flow uniformly, the fluxion of the curve AE shdl
coincide with the velocity by which it is described in a void
(by art, 441), so that we may suppose E/ equal to EK; and
gSa shall be to EN as Z;s is to ER, or the resistance shall be
always as the rectangle contained by the right lines that mea-
sure the fluxions of the curvQ AE .and of SZ ; and this coinddes
with prop. 23, pw. 2, Dcseripi% curv.

453. Because ER is to Zz as the square of E/ is to the recto-
angle contained by £S<i and EN, and the square of EL is to
the square of ^ as 3Z is to Sa^ ER is to Zz as the square of
£ L to the rectangle contained by SZ and EN. Therefore the

density

380 efU^TUmtancsiMi'^ 'BdiOfcT.

ifeoafy (Mth M 9B £R directly^ and tiie ic(ti«re «f ELinverse-
J^)i0 as 2a dnrecUj, and the fccUngle<M>Atoiiied by SZ und
JSN lAT^isdy ; «owcx)(MnlAy> if fhe cmrre A£M be extended
into • i%fat inie, and the ordinate at E be ahrays equal to SZ>
jhe dffpfity at Eahali be always inversely as the sabtangent of

llua figoA.

454* Lrt A&M be any trajeelory tliai can be described m a
vcid by a eentripetal force that is inversely as any power of the
dKotaaaa 8S^ and let it be described in a medium by a centri-
ftetal foroa that is likewise inversely as some power of the
shstaKoe SE ; let 8T pevpendicular to SE meet the tangent in
T^ and the df nsi^ at E shall be always inversely as the tangent
XT. 9br lel BJk and SK be inversely as the powers of the
distMca wiiase egcpoBMts are n and m respectively ; then S2
riialba ahrayi as the power of the distance whose exponent is
m «m, and 2b the flnsioft of SZ i^afl be to the fluxion of
SB in the ratio oomponndad of that of SZ to SE and that df
m-^^ to mi^ by ait. 16? ) therefore fStie density shad be in-
iKrsaly as a ri^t line that is to SE as the fluxion of the cutre
AE is to tbo flaxioftof the toy S£> thai is inversely as the tan-
fltatET.

4W. Thsitiistace nt E ia to die centripetal force in the me-
t^sa at Eas thataetangle contained by one half of E& and 21s;
^ to the rectanglo contained by EN and SZ^ or (because Eb
•I to SF as the floxion of SE to the fluxion of SP^ by art. SB4)
in the ratio compouhdtd 6f the ratios of SP to fiSZj of PE to
f^ and of Za to the fluxion of SP. For example^ if the tra-
|acteiy hethe logarithmic spiral^ and EK the centripetal force in
the medinm be inversely as the power of the distance SE whose
fxpcnent is m ; then^ since Ek is inversely as the cube of the
iUstanee SE, SZ must be as the power of SP whose exponent is
9 — m, the fluxion of SZ aciust be to the fluxion of SP in the
ratio eomppunded of diat of SZ to SP^ and that of 5->m
to unitbiy art. 167 ; and theresistance tothecentripetalforce in
the ratio ccpikpbunded of that of JPE to SE and that of 3— m
to unit. The body cannot descend towards the centre S in this
^iral unless 5 be greater than m; and if it ascend in the spiral,
m must he greater than 9, because SZ must decrease in both

cases;

ChA^i XL XkMJfy ^ <ic MO^m. Mf

eaats^ and teranse SB k to SP hti aa itaMMb^ TiM

deiifit&y^ of the medium b iavttKlf anST «r tte^ktaim^fioitl
S ; aftis shoimby Sir imc Nmfbm^Ptmoip. Mi. % jMp. W»

II we sttppoiethetriftQiQfytQ btowttf thoMooMmcsitdittw^
d9fiorS93 O^g. 170^uid 171)^ and rl>esap{)posed equal to

in thft fonaer^ or to. 9+*;;^ ^ Iattitf> SZl^wU boat tiie

power of the dittanoe wbote ^Xfonmit id r^^fH (bjr art. 452),
endSPaitl^e piMfMr ofthmttstfloiee whe^ eitpMketit itonehalf
of r— 1 . Tbereftite (by ar«. l67)i the t«sbt&Me at atty point
^LsbaUbe to dieutKtripetal ftnm ^lAn the mtio cottpoiittded
of th^ of LP to St aad than of r^-^m \Xf t^L Tbebody
eaattot approach to S in My of those fignfes ttnless r be
greater thaiiiii, audit cwmA reeedefroeiSffi attyofAettiiititesa
r be lets than m. The reaMtanee and' density vatmh tft' the
^ipiiB A io theae AgosM, or in any of those that can be described
in a Yoid by 9^ foroe that is as eny power of the distance. It
was showa above (ttt. 4fi9)> that ihe ratio of the vebcity m
the curve to the vdoctty in a cirde described in a void at the
tame diatanoeby the same centripetal force isBhrays the same
$if when the curve is described in a void.

456* If tfaecentripetal forceaot inpuraOel lities^ let ADCfig. 1 97)
perpendicular to those lines flow unilbrmly; and let DZ be now
to a given right line Da as £K, which measnies the centripetal
feroe in the medium at £^ is to Ek that measures ihe centripetal
ibroe in the void at E or the second fluxion of the ordinate
DE(by art. 418)^ and, the rest xemaining, the resistance ER
will be to Zz as EN is to sDa. If the centripetal force in the
medium be supposed uniform, the rectangle contained by DZ
and Eft must be invariable, and the fluxion of DZ to the flux-
ion of Ek (or the third fluxion of the ordinate DE) as DZ is
to Ek (by prop S), or as the rectangle contained by EK and
Da is to the square of Ek* Therefore the resistance ER is to
Ae gravity EK as the rectangle contained by the right linea
that measui^ the fluxion of the curve AE and the third fluxion
of the ordinate DE is to twice the 3(]^Qare of Eft that measures'

the

58S Of the Re$i$iiuiee and Book 1.

the seoDtid fluxion of the same ordinate* Hie Telocity at E in
the medium b to the velocily at E ia the void in the sahdapli*
cateratioofEKtoEft; and the density of the medium at E is
jedprocaily as a right fine that is to Ei as the fluxion of the
curve is to the fluxion of Ek, or as the right Une that measures
the third fluxion of theordinate DE directly^ and the rectangle
<x)htainedby the right lines that measure the fluxion of thecurve
and the second fluxion of DE invertely. By these theorema
the resistance and density of the medium may be computed vhen
the nature of the curve is known ; but they may be represented
geometricaDy in the following general manner. . ' . •
457- Suppose^asinart.S960^g%149andl50)^thattherectangIe
MTKisalwaysequal to the square of ETj that BV the tangent of
thecurve BKFatB meets ET the tangent of EM in V; then^if
the curve be described by an uniform gravity that acts always
in lines parallel to EB, in a medium whose resistance is as its
density and the square of the velocity of the body together^ the
resistance at E shall be to the gravity as SEB is to 4EVj and
thedensity of themedium shall be always inverselyas the tangent
E V. But if the resistance be supposed to be as the density and
velocity together, let the angle EVu be made equal to EBY on
the same side of EV, and Vu meet EB in u ; then the densitjf
shall beinversely as ^EUTCfig. 158). Letthefigute^forexample^he
any conic section, O the centre, EG a chord in the direction
of the gravity; let the angle EOA: be made equal to GET, and
Ok meet EG in k, and the resistance shall be to the gravity aa
SOk is to SOE ; and if the tangent at E meet thesemidiameter
that bisects EG in V, the density, shall be inversely as £V :.
for, since the triangles EV6, 0£/c are similar (by the 5th pro-
perty of the circle of curvature, art, 375), E6 is to EV as Ok
toOE. When thesectionisacircle,OA: (^^^. 154)becomespe^pteIl-
dicula^ to E6. When the section is an hyperbola, and EB isparal-
lel to one of the asymptotes, the tangents BV and ET intersect
each other in that asymptote • at V, and the resistance is to the
gravity as 3EV is to 2AV. When the tangent in any figure
becomes perpendicular to EB the direction in which the gravity
acts, the ratio of the resistance to half the gravity is the same.
ratio which Sir Isaac Newton calls the index of the variation

of

Chop. XL DenriJhf fifths M^um. S8S

of cttrvature^or the ratio of the fluxion of Ihe ray of curvature
to die fluxioD of the cunre. In oomputing the resistance of the
medium from the second and third fluxions of the ordinate^ or
the right lines ihsX represent them^ regard must be had to what
was shown of these fluxions in chap. 4^ and art. 9M, in order
to avoid mistakes like to those described in art. 440.

458. When the gravi^ acts in parallel lines^ and is either
uniform or variesas anjrpower of the distance from a given plane
G H^ and AEH (fig . 1 98) is any trajectory that could be described
in a void by a force that is iJso as any power of the distance
from GH> the density of the medium at £ is always inversely
as the tangent £T terminated bjr the point of contact £ and by
GH in T. The d^femonstration is similar to that of art. 454.

459- Let AEH be any trajectory described in a void^ or in a
medium that has no resistance^ by a force that acts always ift
right lines (as £M) perpendicular to 6H and in the plane of
the trajectory. Upon £Q that is parallel to GH take QN
so as'always to represent the force at the distance CQ (or EM)
from GH ; let AD represent the force at the dbtance CA, and
the velocity at A be such as would be acquired by falling from m
y^ A with an uniform gravity equal to AD ; complete the rect*
angle AD Ja ; and the velocity at £ shall be to the velocity at
A in the subduplicate ratio of the area aQND/2 to the rectangle
aD. Let AK and £P paraUel to GH and equal to each other
represent the constant fluxion of QE ; let KI and PR meet tl^
tangents AI and ER in I and R ; let ai be to a A as the square
of AK to the square of AI^ and bg parallel to AD meet 'Dd in
g : then ER shall be to AI as the velocity at £ to the velo-
city at A^ or in the subduplicate ratio of aQN(2 to aJ) ; con-
sequently^ PR is to EP^ or the fluxion of the ordinate EM to
the fluxion of the base CM^ in the subduplicate ratio of the
area bQSDg to the rectangle ag ; and hence the construction
of these trajectories may be deduced when the law of the force
is given. For example, when the force is inversely as the cube
of the distance from GH, the trajectory is a conic section.
When the force is as the distance from GH, let A (fig. 199)
be the apsis of the trajectory ; take QE always in the same ra-
tio to the arch AL whose cosine is CQ (CA being radius) in

• ' • ' the.

lb9fldMD^tte«l»ntf<'ofaAtooiieltelfdfCA; tlieaE aball
lie in the thgectory^ add the Mdy^^n ttfovefrom Ato the
llgbt liM CH alwag^ in the sMM time. When the force is
ilrvtmfy m-^tut sqwure of the distiaeeftom GH C^g. 198)> if
Gi^' be^iidto fiCAytUe tin^ectoiy Ib a semicobical parabolt
that has it^obspkat H ii6 thatCH i6to a Btean propoitkmal
iMWin^GA and ab as'Sitto S ; BSbA the cube of £M is equal
tb tile iOtid eoiktained by the sqiMM of HM and a third pro-
^IsktieMi^to ab and fCA; It is constructed by the area of the
drele when C( is less than 2CA> attd by an hyp^bolic area
Hhen Cb is greater than 2CA. In general^ QE is always as
1^ time in which a^ body wouM describe AQ by falling frcmt
A t(^ Q by the same foiroes directed towards C.

460.ThesineoftheangIeMER(>%.l98)istothe8ineofCAlitt
Aesameratio^whentbeveldcityatAwiihthedistanceAQofthe
parallels AK and QB is gifen, and th^ force perpendicular to
OH or QE is the saasie at tbe'same distance from QE. For Af
18 to AK as the radius to the sine of the angle AIK^ or 'CAT,
and £K is to EP as the radius to the sine of the angle M£R;
eensequently^ the sine of the angle MER is to the sine of CAI
as AI to ER^ or in the subdupDcate ratio of aD to the arte
€tQSid; but this ratio remains the same in all the different po-
sitions of the tangent AI : therefore^ when a ray of light passes
through any given medium tbatactsupon itin parallel llnesper^
pendicular to the planes that terminate the medium^ the sinecf
the angle of refraction is to the sine of the angle of incidence
in an invariable ratfo^ and the velocity with which the ray
emerges at E is to the velocity of itsincidenceatAin the inverse
ratio; asSir Jsaac Nearfon has shown in adifferentmanner^prup.
94 and 95, lib, I, Princip. If we suppose (fig. 20Cf) the ray to
move from E towards A^ and the angle MER to be increased
till its sine be to die radius in the subduplicate ratio of aD to
aQJid, the angle CAI will become a right one, or A will be the
apsis of the trajectory, and die ray will be reflected from A,
so as, after returning to the right line EQe at e, to emerge in
an angle equal to the angle of incidence MER.

461. When the body is projected from A in a right line AI
that is not in the plane ACH, the force bemg still p«fp^dt*

cular

Chap. Xh toWafA ttoetal CdShrtu 88K

cular to CH^ l6iAI represent the vdocity of th^f>rojectioo, and
IV parallel to CH meet a plane AKB that is perpcndicalar to
CH in V; and the same tngectorj will be described as when the
body is project^ in the phne AKB with the Tdocity AV, and
gravitates towards C by a force that varies in the same maaner
as the force towards the axis CH> while the plane AKBC isat
the same time carried along' CH with the Telocity IV.

462. Hitherto wehave supposed the farces to aciia ri^t linei
that meet in one given point> or are perpendicalar to one gtved
right line. Let us now suppose that centripetal forces are di>
rected towards any number of fixed points^ and that die foh^
towards each is always the same at the same djstanoe Afom itt
then if abody move fromAtoE^Cj(g.9Ol)>ilttTelocityai£0baU
beihesamewhetheritmoveinanycurveMnfiAZEorinthcchanl
A£^ if the velocity at A be the same in both cases. For bf
whatwes shown in art. 435, the incitement or decrement of the
velocity generated by the force directed towards any <me mb^
tre C is the same whether the body move in a curve or in a
light line from A to £> if it set out from A with the same velof
city; and when any number of farces directed towards severtf
centres act upon the body, the square of the velocity at £ {«
measured by the aggregate of the same areas, wh^er^ihe bodjr
move in the curve AZE or chord AE. •

463« Let AZE be such a surface that when a body is placed
upon it the actions of all the centres towards which it is attract*
ed balance each other, and the body issustamedby thesarfaceaa
as to move neither way; then if a body attracted towards those
centres in the s^one manner move in any line that meets tl^s
surface in A and E, its velocities at these points shall be equal;
and if a body move from any given point with a gfven velocitji^
and arrive at this surface, its velocity when it comes to this sur*
face will be always the same. Hence the proper resolution is
deduced of a project for a perpetual movement mentioned by a
celebrated Author. AloadstoneatA (/g.202)issnpposed tohave
a sufficient force tobring up a heavy body along the planeFA from
F to B, whence the body is supposed to descend by its gravity
along the curve BEF till it return to its first place F, and thus
to rise along tibe plane FA, and descend along the curve BEF

VOL. L B b continually.

880 CfGrmitaiidn Bookie

oontiitiiiilly* But supposing BZE to be ihe surface upon which
if a body was placed the attraction of the loadstone and the gra*
▼ily of the body would balance each other, this surface shall
meet BEF lit some point £ betwixt A and F^and the body must
stop in descenditig along AEF at the point E. See tVilkin^^
Mothem. magk Book 11. Ckap.lS.

464. Let AZE(;£g.£Ol)beatfajectory described by forces di«
reeted towards the centres CandS> thesquareof theyetocity at A
be represented by the rectangle dD, and the forces at A towards
C and S by the ordinates AD and Ad perpendicular to CA and
SA, respectively; let CQ and Sf betaken upon CAand SAie-
q>ectiyely equal to C£ and SE, and the ordinates QNj qn al*
ways represent the forces towards C andS at the distances CB
and SE; then if the square of V be equal to twice the ag*
gvegate of the areas aD, AQND, and Aqnd joined with their
proper signs, V shall measure the velocity at E. And in the
same manner V is determined when there are more centres that
attract the body* Let EH be taken upon EC, and EL upon £S
respectively equal to QN and qn^ complete the parallelogram
BHKL, then die force at E that results from the attraction of
the several centres shall act in the direction EK and be measur*
ed by EK. Let the circle of curvature at E meet EK in B, hi*
sect EB in b, and the rectangle £EK shall beequaltothe square
of W, by art. 440. And if KR parallel to the tangent at E
meet any other chord of the circle of curvature, as ED, in R,
and ED be bisected in d, the rectangle dER shall be equal to
the square of V; for the angle ERK is equal to EBD, EB to
ED as ER to EK, and the rectangle cfER equal to &EK«
Hence^ if KP be perpendicular to the tangent at E in P, the
rectangle contained by KP and the ray of curvature is equal to
the square of V ; and if KR meet T£ (that is drawn from E to
any fixed point T) in x, the rectangle contained by Ex and half
the chord of curvature that passes through T shidl be equal to
the square of V.

465. Let T(/g.203) be any fixed point, TC the base of the
figure, and T A perpendicular toTC;Iet EM the ordinate from £
meet the base in M, and EN parallel to the base meet TA in N;
let KH parallel to TA meet EM in H ; then the time being al«

ways

I

Chdp. XI. UmafSi Mtiferal Centres, sVf

ways supposed to flow unifbnnljr, and the flttzk>n of Cb6 carve
being represented by V which measures ^e velocity at £^ the .
second fluxion of the ordinate shall be always measured by.
KH^ and the second fluxion of the base by £H^ or KL: that is^
if while E describes the triyectoiy^ £M and £N be always;
perpendicular to TC. and TA in M and N^ the powers by.
which the motions of the points M and N are accelerated or
retarded shall be measured by £L and £H» For let Et be
taken upon the tangrat equal to V^ and tl parallel to TA meet
£N in I, that Bi, El, and U may measure the fluxions of the
curve, base, and ordinate, respectivdy. Let ^t measure the
aecond fluxion of the ordinate, and supposing O to be the
centre of curvature, let Om be perpendicular to £M; in m ;
and KP being perpendicular to £^, let Pz be perpendicular to
£N in z. Then the second fluxion of the curve (or the power,
that accelerates the motion of £) shall be m^eaaiured by £P «
The rectangle contained by Ofi and b b equal to that which
is contained by Om and Et, and (because the fluxions of those
rectangles are equal, and while C£ increases. Om and It
decrease) the rectangle contained by £1 and £/ is equal to the
sum of the two rectangles contained by Om and £P, 4nd>y
0£ and it. By the last article, KP is to £^ (or V) as £t is tof
0£; and the sum of KH and P;r is to KP as £1 to Et; there-
fore the sum of the two rectangles contained by OE and KH,
and by 0£ and Fz, is equal to the rectangle contained by £1
and Et, or to the sum of the two rectangles contained by OE
and tt and by Qm and EP ; but (because of the similar trian-*
gles OmE and PzE) the rectangle contained by 0£ and Px
is equal to that which is contained by Om and EP ; conse^
quently, the rectangle contained by ()£ and KH is equal to
the rectangle contained by OE and tt, and KH is equal to tt
which was supposed to measure the second fluxion of the or*
dinate £M. In the same manner it appears that KL, or £H,
measures the second fluxion of the base TM. This theorem
holds, not only when the angle MTN is right, but when MTN
TME and TNE are any given angles, providing KL parallel
io TC meet EM in L, and KH parallel to TA meet EN in
H ; for KH will always measure the second fluxion of TN^
and KL the second fluxion of TM.

BbS 466.1111

3nr Of tiraviiaiion ' fiook T«

468. Iliis priilclplej however, that when the force which re^
salts from the attraction of all the centres is measured by £K^
and is resolved into the forces EH and EL parallel to TA and
TC respectively, the second fluxions of the ordinate £M and
base TM are measured by EH and EL respectively, may be
admitted without this proof* For, if we resolve the motion at
£ in the direction Ei into a motion Elin a right fine parallel to
TC, and a motion £i pardilel to TA, we may conceive the body
tb i^eBCend in the right line £N with a velocity EI that is ac-
celerated by the force EH, while the right line EN moves
parUlel to itself i^ng TA with a velocity Et that is retarded
by the force EL; and thus it may appear (as in art* 418), that
dse force £H measures the secdnd fluxion of EN, and \bt
force EL the second fluxion of EM. The position of EH and
£L serve likewise to riiow whether these second finxionB are to
be considered asal&rmaliveor negative: thus when TM decreases
and EM increasi^ and £K b mthin Urn angle M£N, the se-
cond fluxionsnof TM and EM are to be both conadered as ne-
gative : but if EK be witliin the angle i£N, the second fluxion
of EM is^to be considered as positive.

467* T^is principle being admitted, several of the theorems
in the preceding articles may be readily demonstrated from it.
Thus if the force EK be directed towards any fixed point that is
any where in -the right line EM, EK will coindde with EL, so.
that EH will vanish ; consequently the centripetal force will be
measured by the second fluxion of the ordinate EM when TM
flows uniformly, as in art 425. When the force is directed
towards the fixed point T, the area desKsribed about T flows
uniformly, or the fluxion of the area CTE is invariable ; for the
itrea CTE is half the sum of the aieas CME and CTNE, the
fluxionsof which are measured by the rectangles IM andiN,by
prop.4. Thefluxion of the rectangle IM ismealuredby thesum
•of the rectangles HM and U, and the fluxion of iN by thesum of
the rectangles LK and li, by prop. 3^ But while CM and ME
increase, EN and It decrease ; consequently the second fluxion
^-of the aiea CTE(or thefluxion of the sum of the rectangles IM
and iN) is measured by the excess of HM and li above LNand
Ji, ibatis^bytheexcessof HMabove LN, or-of KM above KN.

Bat

fShnp. XL towards tevcrai Ctnatn. 86§

Sot when EK {lasaes throngh T^ this excess vanishes^ md thf
area described about T flows umformly ; which is the first prot
position of the first book of Sir Isaac Newton's p^nciplesi
and was demonstrated after his method above in art* 441, an4
in a manner that differs little from Mn Herman\ in art. 442.
, 468. In general it appears that the second fluxion of the area
CT£ is measured by the excess of the rectangle KM above KN^
or (if EK meet TA in V) by the rectangle contained Vy KL
and TV ; and the fluxion of the area CTE increases or decreases
according as EK is on the side of £T towards which the body
moves^orontheoppositeside. The theorem in art. 431 0fg.£03>
n. 2) may be likewise made more general ; for let Eube acircle de»
scribed about the centre T^ and the uniform angular velocity of
Tu be equal to the angular velocity of TE while £ moves in thf
trajectory. Let tip parallel to TE meet the trajectory in x, and
meet Tp perpendicular to T£ in p ; and let e move in the right
line TE so that Te be always equal to Tjt : then (by art. S85X
the secondfluxion of Te (or Tx) shall be equal to the difference
of the second fluxions of px and pu when x and u set out toge»
ther from E. But if K^ be perpendicular to TE in y^ E^ shall
measure the second fluxion oipx at that term ; and the force by
which the circle Eti is described about T^ is measured by the
second fluxion of pu at the same term^ by art. 422. From
which it follows^ that the force by which the motion of e is
accelerated or retarded^ when it sets out from E^ is equal to the
difierence of the force Ey and the force by which the circle Ey
is described about T^ or the centrifugal force that arises froqi
the circulatory motion of E about T. Of this principle^ see
the Lqwsof the Moon^smotion according to gravity, p. 65 and 66.
469* We supposed the trajectory described by.E to be in one
plane in art. 466 and 467 • But similai conclusions are easily de*
duced when the trajectory cDe is no t in one plane. In this case let
CE (fig. 204)be theorthographic projection of the trajectory on
the plane CTA, or eE be always perpendicular from the trajectory
to CTA in E^ and EK be the projection of the right line ek
which represents the force that results from the action of the
several centres upon jthe body at e with its direction. Then the
^cond fluxions of £M and EN sl^^m be measured by EL aiivt

Bb3 EH;

1

SgO Of Grteoitation^ Book T.

£H ; and the second fluxion of eE^ the distance of the body
from the plane CTA shall be measured by the difference of the
l^rpendicolars ftK and « E. From which it follows^ that 'when
a tnyectory is described by a force that is directed towards a
fixed point T> and another force that is always perpendicnlar
to the plane CTA^ the area described by TE about T on this
plane flows uniformly. '

470.LetabodysetoutfromagivenpointD(^g.£04)withagiveii
▼docity^ and first let the trajectory be described in the plane
CTA. Let the force at D that results from the attraction of the
aevieral centres be represented by Tif, and be. resolved into D/
and DA parallel to TA and TC, respectively. Let DP and DG
be perpendicular to TCand TA in F and G ; upon which pro-
duced take F^equal to Jih, and Qg equal to D/. Let Mm and
N» betaken in thesame manner upon EM and EN always equal
to EH and EL^ respectively. Let the velocity at D in the di-
leotion DO parallel to CT be such as would be acquired by a
body falling from B to F with an uniform gravity equal to the
force Tf; and let the velocity at D in the direction parallel to
TA be such as would be acquired in the same manner by a body
lalling from A to G with an uniform gravity equal to the force
Gg ; or^ in general^ let these velocities be such as that half the
square of each may be measured by the rectangles Yh and Ga,
according to art. 434. Suppose likewise that while the body
moves from D to E^TM decreases^ but that EM increases^ and
that EK falls within the angle MEN. Then the square of the
velocity of M (or of the fluxion of the base TM) shall be mea*
sured by iBlAmfb, and the square of the velocity of N (or of
the fluxion of the ordinate EM) by %Ga — sGNitf . For let
My measure the velocity of M^^ or the fluxion of CM, and the
fluxion of this velocity (or second fluxbn of CM) being mea?
sured by EH or Mm (by art. 465), it follows that the fluxion
of the square of Mj isequal to the fluxion of2BMi»/%,and that
the square of My is equal to 2BMmf 6. In the same manner the
square of the velocity of N is measured by 9Ga — fiGN/i^.
And the fluxion of the base TM is to the fluxion of the ordinate
EMinthesubduphcateratiooftheareaBMiTi/'itoGa — GNitg.
When the trajectory cDt b ROt in the plane CTAj^ the motions

at

Gfaap. XL towards stoeral Ceritra. SOd

at D and E are'each to be resolved into three motioiis in tbe^
rectioQS parallel to CT, TA, and that which k perpendicnlar
toCTA; the forces at D and E are each to be resolved likewise
into three forces in these directions ; and the flaxions of TM^
ME^ and £« are to be determined in the same manner as the
fluxions of TM and ME in the former case^

47 1 . There arises hence a simple enongh constmction of the
trajectorythatwouldbe described by the moon about the earth,
if^thegravity towards the sunbeinginverseiy as the squareof the
distance, the gravity towards the earth varied in thesamepropOF*
tion as the distance from its centre, and we should abstract from
the curvature of the earth's orbit during a revohitioii of the
moon; which weshall subjoin, becauseitserves to iUustiatesome
parts of the theory of the moon's motion. For though the gm*
vity towards the earth be not as the distance from ita centre,
but inversely as the square of that distance ; yetwben the osfait
of the moon is supposed to approach nearly to a cird^ some of
the.efiects of the solar action deduced from these suppositioiis
will nearlycoincide, and the trajectory ismoreeasily determined
in the former case tlian in the latter. LetS C^.205)repre8eiitdie
sun, T the earth, CADB the orbit of the moon about the earth,
C and D the quadratures, A the conjunction, Bthe opposition,
E the moon's place in this orbit at any time, and EN a perpen*
dicular from E onSTthat joins the centresof the sun and earth.
If the sun acted on the earth and moon with equal forces, and
in parallel lines, they would fall equally towards the sun in pa-^
rallel lines, and the solaraction would not disturbthe motioned
the moon and earth about their common centreofgravity,orthe
motion of the moon about the earth, when the motion is allre*
ferred to themoon,asisusual. Butsincethesnnactswithgieat*
er force upon the moon at A than upon the earth at T, and Hp«
on the e&rth at T with more force than upon the moon at B ;
it follows that, if the earth and moon fell towards the sun, the
moon would fall more than the earth toward^ the sun in the
former case, and the earth more than the moon in the latter ;
so that their distance from each other wouldincrease by the in-
equality of the solar action in both cases. If the moon fell
either from C or D towards, the sun, and the earth from T, their

B b 4 distance

diflbiabe Iromeaehoiherwoiilddecreaseby their fallinginrigjht
lioea that meet in the same centre S. And it is obvious that iha
action of the sun mustdiBunish the force by which they tett4
toirafds each other where it would increase their distance if
they fell towards the sun j that is a^ A and B; but must increase
that forcewhere it would caose them to approach to eacbother^
that fi at C and D.

. 47&» Let ST represent the gravity of the earth towards tfa«
sun^ and if S^ be taken upon SS in the same proportion to ST
as the square of ST is to the square of S£> ^wiU represent
the gravity of the moon at £ towards the sun. Let^ parallel
to ET aeet ST in g, and the force S^being resolved into $g
(which acts at £ in a right line parallel to TS) and /jg which
^Msti in the directjon ET, Sir Isaac NifiptM n^lects the part
ST pf the force Sg, because it is equal to the gravity of the
earth towards the sun* and acts in a parallel line ; and* because
of the vast distance of the sun* he couiuders ^as equal to £T
(the rather that as^ exceeds £T when £ is in the part of tho
sttoon's orbit CAD* it is less than £T in the part BBC* and the
exdess in the former is nearly equal to the defect in the latter
case when tfae angle CT£ is ^ same); and* Sf being supposed
to meet CD in A* f^ is nearly double of £A* and Tg nearly
aqoid to dTN. If TN be to N» as the square of ST is to tbe
square of TN* tbe mean value of Tg will be more nearly equal
to STN4-IO Nn ; and a construction of the tr^ectory may be
derived from the preceding articles upon this supposition like-
wise ; but as this construction is mor^ complex, we shall supt
pose hjere (with Sir Isaac Newion) that Tg is equal to 3TN ;
so that the trajectory may be supposed to be described by
these three forces^ the gravity of the moon towards the earth,
a force directed towards tbe earth that is to the gravity of thQ
earth towards the sun as £T the distance of the earth andmooq
is to ST the distance of the earth and sun* and a third force ^
that acts in alright line parallel to TS* which is to the second
force as STN is tp ^T. I^t G represent the gravity of the
^apon towards the earth at the distance TC* V the force^
which the solar action adds to this gravity when the moon is ii;
quadrature to the sun at C| S tlie periodic time of (be earth

about

Chip* XI. towards teperal Cenires^ 999

about the wn, L the periodic time of the moofit about dieeaitii»
I the time in which a circle would be described about T at Ifae
distance TC by G-f- V the whole force at C towards T, and.
I the time in which a circle would be described at the same
distance by the force G only. Then^ by art 459, 6 the gravity
^the moon towards the earth is to the gravity of the earth to*
wards the sun in thelratio compounded of the ratio of TC to
ST and of SS to U : and the gravity of the earth towards the
sun is to V as ST is to TC; consequently G is to V asSS is to
)I. By the same article^ II is to /^ as G-f V is to G^ and
therefore SS is to H &sG4-V is to V. This being premised,
the trajectory that would be described by the moon, if thQ
gravity towards the earth was varied in the same proportiou
as the distance, and we should abstract from the angular mo*
' tion of the right line TS, may be constructed in the following
laanner.

473. Firstlettbe moonbesupposed to setoutfromli(^g.£06)
In any direction LZ that is in the plane LTS ; let the velocity
of the motion be represented by LZ and be resolved into LY
parallel to CT and YZ parallel to TS; and let LF and LG be
perpendicular to TC and TA in F and G. Upon the right line
TC take TB the distance froQi T by falling from which to F ^
Telocity would be acquired at F equal to LY, and TA the
distance by falling from which to 6 in the right line AG a
velocity would be acquired at G equal to YZ. From the centre
T describe the circle BH& meeting FL in H, and the circle
Ala meeting G L in I^ and join TH and TL Then in order to
construct the trajectory draw any right line TP meeting the
circle BH6 in P, draw TQ so that the angle ITQ may be alwajt
to the angle HTP in the subduplicate ratio of G — SV to G-|-V^
nd let TQ meet the circle Ala in Q* Then PM a right lin^
through P parallel to TA shall always intersect QN aright line
through Q parallel to BT in E a point of the trajectory : and
the a^k L£ of thi^ trajectory shall be described in the same time
that the ark HP is described by a body revolving with an uni-
form motion in the circle B^b by a force that is to G-f-Y as
TB is to TC. This construction is deduced from art. 470«
fjcom which it follpws that if Bb represent the whole force at

. Bto-

594 Of Gravifafion Sock f .

B toirards T and Tb meet PM in m, Aa the fiocce at A to*
wtf ds T and Ta meet QN in n, the flaxioa of the base T&f
ibali be to the fluxion of the ordinate ME (or TN) in the sub*
da|>}rcate ratioof the trapezium BMfitb to the trapezinm ANna,
fhat 18^ in the ratio compounded of that of PM to QN, and of
the subduplicate ratio of the fc^rces at equal distances from T in

the right Tines TC and TS, or ofv'G+V to •G--2V. The
fluxion of the angle HTP is to the fluxion of ITQ in the ratio
compounded of that of the fluxion of TM to the fluxion of TN,
and of that of QN to PM. Therefore the fluxion of the angle

HTP is to the fluxion of ITQ as •G+V to VG— «V ; and
these angles themselves are in the same ratio^by art.24. Whence
the construction is manifest. We suppose V to be le&s than 6
in this construction. If V was equal to G, the trajectory would
be the same that was constructed in art. 459- when the gravity
was supposed to act in right lines perpendicular to a given right
Kne; and to be as the distance from it. If V was greater than Gj
the construction would depend on the' logarithms, or on hy-
perbolic area% but it would be of no use for our present
purpose to describe that case*

474. The fluxion of TM is to the fluxion of HP, or the ve-
locity of the point M to the velocity of P, as PM to TB : but
the velocity of M is as PM ; therefore the velocity of Pis as TB,
so that its motion isuniform,andis the same with which the circle
BHP is described by a force directed towards T that is to G + V
as TB is to TC. The motion of Q is likewise uniform, and
is the same with which tlie circle IQA is described when the
centripetal force is directed towards T, and is to G — fiV as
TA to TC. And the arch LE is described in the trajectory in
the same time that P with its uniform motion describes HP, or
Q describes IQ. Let TK be taken upon TA in the same ratio
to TN as S V is tp G +'V. The gravity at E towards T is to
G as ET to TC, by our supposition; the force added by the
solar action at E is to V in the same ratio: and the sum of these
forces is to G+V in that ratio, likewise. The third force Eft
(d^ribed in art. 472) is to V as 3 TN to TC, and to G+V as '
Tk to TC; therefore the force E* is to the sum of the oth^r
two forces asTK to £T; consequently the force that results

from

Rate TSXm- fb.,3ff4Si>i. I .

Tfy.jro^jTfx.

>

I "Tf

'T*

/{

w/^

/

."^'

\

<" ™

\

S

1

Chap. XI. towards several Centres. Sg5

firom the three forces acts in the direction £K^ and is to G'^V
as EK to TC.

475. Let Pr be perpendi cular t o TA in r, upon rP take rR
to QN as •G— 2V is to vG+VJoin TR^ and let Ei be per-
pendicular to TR from E ; then Et shall be the tangent of the
trajectory at E, and the velocity at E shall be to the constant
Tolocity of P as TR is to TB. For the fluxion of TM is to
the fluxion of TN as PM (or Tr) to rR (by what was shown
in art. 473) and to the fluxion of the curve LE as Tr to TR,
therefore the tangent intersects EN in an angle equal to RTr^
(by prop. 14), and is perpendicular to TR. The velocity at E
is to the velocity of M as TR to Tr ; the velocity of M is to
the velocity of P as Tr to TP; consequently the velocity at E
is to the constant velocity of P as TR to TB.

476. * Let the body now set out from any point L (fig* 207)
in a right line L/that is not in the plane LTS, audits motion
being represented by hf, let it be resolved into a motion /Z la
a direction perpendicular to the plane LTS and a motion
LZ in that plane, and let the latter be resolved as above into
two motions in the directions LY and YZ represented by these
right lines. Then tlie rest of the construction being the same
as in art. 47 3, let the motion /Z be such as ^ body would acquire
by falling from D to T by the force that acts in the line of the
quadratures BT. Let a circle Dd described from the centre T
with the radius TD meet TP in X, from which let XV be per-
pendicular to TH in V; and the point E being determined as
in art. 473. if E« be perpendicular to the plane LTS at £ and
equal to XV, e shall be a point in the trajectory. When the
body sets out from any point that is not in the plane LTS^^
the points of the trajectory are determined by a similar coa«*
struction. The point L is one of the nodes of the trajectoxy^
or of the points wherein it intersects the plane LTS ; and since
XV vanishes when HP becomes equal to a semicircle or to any
multiple of a semicircle, by the construction, it follows that
the revolving body returns to the plane. LTS every time the
point P comes to theright line HTA; or that £ becomes a node
of the trajectory, and TE the line of the nodes, when HP be-
oomes equal to a semicircle or to any multiple of a semlcirole.

477. If

SgO .' OfGrmiiation ' Sook i^

477« If a rectangle DG be described so as to touch thm
circle Aa in the opposite points A and Oy and the circle Bi n%
the opposite points B and b, it appears from the constnictioii
that the cvrve described by the point £ (which is the projection
ef the trajectory on the plane LTS) will touch the four 8ide»
of this rectangle in every revolution. The points in which it
looches the sides Gg, Dd that are perpendicular to Bi move
backwards, and the points in which it touches the sides dG and
gD that are perpendicular to Aa move forwards^ till tliey come
to the angles G and D. The angles HTP, ITQ and PTQ in-
crease uniformly ; and when PTQ becomes equal to a right one
if TP iftlla upon TA, the point E will pass through T, and the
ssvolving body will pass above T at a distance that is to TD. ai^
the sine of the angle ATH is to the radius. Thus if H coincide
with B, or the body set out fircwn the plane L TS (Jg. 208) at
ilie qo adratore B, and the excess of -/G+V above ^G — 2V
he to yO+V as unit is to any integer number m^ the angle PTQ
wiU be to BTP in the same ratio^ PTQ will become a righ^
angle after TP has described as many right ones as there are
mits in m, the point £ will then pass through T^ and the per-
pendicular £<will beequal toTD, The points in which the curve
described by E touch Gg and Dd (which are in this case the
nedes of the trstjectory^ or the points in which it intersects the
plane LTS) in this time move with a retrograde motion from B
to G and A to D. Aftor this the motion of E becomes retro*
grade^the motion of the points where the curve LE touches Gg
and Dd (or of the nodes of the trajectory) becomes direct; but
the motion of the points where it touches dG andgD is then re-
trograde, and has always the same direction with the motion of
E. And when TP has described as many right angles as there
aie units in 4fn, the revolving body returns to B with its first
velocity and direction. The inclination of the plane of the tra-
jectory first increases, till it become perpendicular to the plane
LTS when TP has describedas many right angles bs there are
units in m ; thereafter it decreases and returns to its first mag-
nitude in an equal time. The revolutions in this trajectory
from any quadrature to the subsequent quadrature on the same
Side of T aie completed in times equal to each other^ and to

the

^ap. Xl. * iowaris several CenHret. ^

the time in wliich the point Q revolves in the circle Aa by. 4'
force that is to G— «V as TA is to TC* The revolatioiis
ftem any conjunction to the subsequent conjunction are com*
pleted in Hie same time that P revolves in the circle B6 by a
force that is to G4-V as T&is to TC; and these times are to
each other in the subduplicate ratio of G-J-V to G — SV*
What has been observed in this article holds only upon the
supposition that the. right line TS has no motion about the
points T or S^ and that the force towards T is as the distance;
for when TS revolves about either of those points^ as in the
case of the secondary planets^ the nodes move backwards Hi
every revolution^ the inclination of the plane increases and de«
creases within narrower limits^ and the odier effects of Git
forces jf^ and T^ are diflerent.

478. It appears from the construction likewi^ that when V
is much less than G, H coincides with B, and TA is nearly
equal to TB^ the distance TE is least not far from the octants
before the syzygies, when TE bisects the angle BTA, and is
greatest after the other octants when TE bisects AT6. After some
revolutions the figure of the trajectory differs little frbm an d-
lipse that has its transverse axis in the latter and the conjugate
axis in the former position; for if the angle QTP was inva-
riable^ TA equal to TB^ and QN parallel to BT intersected
PM parallel to AT always in E (as in this construction), the
curve described by E would be an ellipse whose transverse a^scis
would bisect the angles AT6 and aTB. The tides produced
in the ocean by the inequality of the gravitation of the watet
towards the moon rise to the greatest heights about the same
octants ; of this see the latter part of CoroL 20, Prop. 66. Lib. I.
Princip.

479* Suppose 4hat the body sets out from the node and qni^^^r
dratttreB(/?g.^8) inarightlineperpendiculartoTB^ and the rest
remaining as in art. 476> itismanifestfrom the construction that
the body will return to this node again, when the ray TP after
completing a revolution about T returns to the situation TB;
in which time the ray TQ describes an angle that is to four
right ones in the subduplicate ratio of G-*-€V to G+V.
Tfaer^re if the angle BT; be constituted on that side of IB

which

598 Of the Motion of tie Plane Book h

which is towardg the opposition a in the same ratio to Ibor
right ones as the excess of i/gTv above •g— 3V is to •o-i-v*
and T; meet the circle Aa in f j then qe be drawn parallel to
TB and meet Be parallel to AT in is, the point e shall be the
node of the trajectory after this revolution of P> and the angle
BTe will show how far the line of the nodes has moved back'*
wards during this revolution* Let the motion at B be resolve
ed into /Z perpendicular to the plane BTS (which represents
the ecliptic) and Bz parallel to TA ^ and if the motion at B be
given* or 1^ (which represents this motion) be invariable, the
motion BZ will vary in the s»ne ratio as the cosine of the angle
/BZ which is the inclination of the plane T^to the ediptic
But the right line TA by falling along which this motion
would be acquired varies in the same proportion as this motioD,
Be is equal to the sine of the angle BT;^ the radius being Ty or
TA, and the angle BTy is given when the ratio of G to V is
given; consequently Be varies in the same ratio as the cosine of
the inclination /BZ; that is the tangent of the angle BTe is as
the cosine of the inclination ; and when V is much less than G,
the angle BTe varies nearly in the same proportion. Therefore
when the velociy at B with the ratio of G to V are given, the
angle T^ is right, and V is much less than G> the retrograde
motion of the node during this revolution is nearly as the oo*
sine of the inclination of the plane.

480. In order to estimate from this construction what ought
to be the motion of the nodes of the moon nearly, we most
suppose this trajectory to be as near to a circle as possible, and
the body to revolve in it in the same time that the moon re-
volves about the earth. First, let the line of the nodes be in
quadrature to the sun about the middle of the month, and the
inclination of the plane be supposed incomparably small; then
TA (fig> 209) being supposedequal to TB (that the orbit may be
nearly circular) and TB equ al to TC,let the time in which the body
Erevolves from the quadrature in TB to the subsequait quadra-
ture in the right line T6beequaltothe time in which the moon
revolves betwixt these quadratures. Then since XV vanishes
when HP becomes equal to a semicircle, it follows that E re-
volves from the node inTB to the subsequent node n in the time

P makes

Cbi^XI. ^f the Lunar OrbU. Sg^

F makes half a revolution about T in the circle Bb ; bvt E re-
volves from the quadrature in TB to the subsequent quadrature
inTiin the time Qmakes half arevolutionaboutT in the circle
Au ; and since the direct motion of the node while £ proceeds
from the right line Tn to the quadrature in the right line Tb is
so exceeding small that it may be neglected (as will appear better
afterwards), it follows that the mean motion of the node is to
the mean motion of the moon in this month as the difference
of the times in which Q and P revolve about T is to the time
in which Q revolves about T, that is as the difierence of
VG+V and VG—^V is to •ij+V. Let S and L (as in art.
473) represent the periodic times of the earth about the sun and
of the moon about the earth ; then since / isequal to the time in
which P revolves about T in the circle B6 by the force G-j-V,
and L is equal to Uie time in which Q revolves about T^
// will be to LL as G— 2V to G+V. But by art 473,
SS is to // as G+V is to V; consequently SS is to LL
as G — 2V is to V; and G + V is to G— sV a$
SS-fSLL is to SS. Therefore when the inclination of the
orbit is incomparably small* and the orbit nearly circular^ the
mean motion of tlie node is la the mean motion of the moon ii|
the month when the line o f the node s is in quadrature to the sua
nearly as the excess of v'i^S+sLLabove S is to •Sb+aLL.
And if there are 2159 revolutions of the moon to the stars ia
l60 sidereal years (and consequently SS to LLin the duplicate
ratio of these numbers), this proportion is that of 1 to 120,647^
By the principles laid down in the excellent treatise concern*-
ingthe Lawsofthe Moon's Motion according togravity^ihispTo*
portion is about that of 1 to 120,639- If we would compare
it with the ratio that results from Sir Isaac Newton's method,
their difference will appear very small if we may adapt his me-
thod to our present supposition in the following manner. It
appears from what was shown above (art. 432), that when a
force acts upon a body in a right line perpendicular to the di*
rection of its motion, the deflexion of its course from a right
line (or the angular velocity of the right line that is the di*
rection of its motion, and is dways the tangent of the trajectory
at the body) is as that force when tlie velocity is given, and in

general

406 Of tit Motion of the Plane Bolik t.

general a& the force directly and velocity mversely. Henoe
(according to hig method) Uie motion of the node when the
ttioon h at the conjunction is to the inflexion of the coone of
the moon from a right line there, as the force that pfo-
dooes the motion of the node at the conjunction to the force
that acts upon the moon there, that is as 3V to G — 2V ; and
thi^ inflexion of the orbit of the moon is to the inflexion of Q
in describing the circle Aa (which Q describes by the same
force G — 2V) as the velocity of Q is to the velocity of the
moon at the conjunc tion, or to the velocity of P in the circle B6,
that is as V^G — OV is to -/G+V ; so that, according to thb
method, the motion of the node at the conjunction A is to the
motion of Q, or the mean motion of the moon, in the ratio
impounded of the ratio of SV to O — 2V, and of the subdo*
j>licate ratio of G — sV to G+V, that is itt the ratio of 3LL
to S i/SS+SLL. According to which proportion the mean
motion of the node in the month when the line of the nodes is
in quadrature to the sun and the plane of the orbit is supposed
almost coincident with the ecliptic, is to the mean motion of
the moon as I to 120^643. And these three proportions agree
nearly, the last of which is almost a mean betwixt the other
two.

481. The mean motion of the node being determined when
the planes are almost coincident, if this motion be diminished
in the ratio of the cosine of the inclination of the plane of the
moon's orbit to the radius^ we shall obtain the mean motion of
the nodes of the moon in this month nearly, by art. 479* If
we diminish the motion of the node that was deduced at
the end of the last article in the ratio of the cosine of
4^59'.35". (the inclination at the sjrzygies in this month) to the
radius^ the mean motion of the node in this month (according
to this method)shalI be to themeanmotionof the moon as 1 to
121,lCK23j and the mean hourly motion of thenode I6'' ig'" ^.
If we diminish the motion of the node deduced from our
construction in the same ratio, the mean motion of the node
will be to the mean motion of the moon as 1 to 121,10648, and
the mean hourly motion of the node will be about 16"19'* i*
The same nourly motion of the node by the principles that are

pro-

ChAp- XL of tii Lunar OrUt. 40T

proposed in the tieatiae of the laws of the fnoon'smotton accord-
itig to gravity ia about l&* 19"' tb« How much the mo(bn of
the Bode amounts to in any of the trajectories constructed ia
art. 476^ in a revolution or any part of a revolution^ will appear
from what follows*

482. Sopposing firstH (fig. 210) to ooincidewith B asin art»
479^ let P/ perpendicular to TP meet TB in I, and In pari^el
toATmeetdietangentat£(whichispeTpendiculartoTRbyartk
473)^ in n join Tn, and it shall be the line of the nodes when
the revolving body comes to e after describing the arch of the
trajectory Be. For supposing the tangent of the trigectory at e
to meet ^E (the tangent of the curve described by B) in any
point n, and nh parallel to BT to meet PM in A ; then because
£e is always equal to XV^ the fluxion of £e shall be to the
fluxion of TV as TV is to XV ; the fluxion of TV is to th^
fluxion of TM as XV is to PM ; therefore the fluxion of Be ia
to the fluxion of TM as TV is to PM ; consequently Be is ta
An as TV to PM : and An is to PM as Be ^orXV) to TV, or
as PMtoTM, or M/ toPM; so that hn is equal to W, and In
parallel to TA ; whence the construction is manifest. Whea
P/ becomes parallel to TB> the line of the nodes becomes pa-
rallel to the tangent at E, or perpendicular to TR. The velo-
city at € in the trajectory is to ihe constant velocity of P in the
mbduplicate ratio of the sum of the squares of TR and TV to
the square of TB. The ratio of TA to TB is compounded of
the ratio of the velocity in the trigectory at B to the velocity
of P, the ratio of the co sine of t he i nclinatio n of the plane to
the radius, and that of V G+ V to •Q w-g V,

483- LetmNbe to QNas •G— 2V to •G+V, andmo pa-
rallel to TP meetP/ perpendicular to TP in 0, then the. angle
PTo shall be equal to BTii, which shows how far the line of
the nodes moves from the quadrature while the arch Be ia
described in the trajectory. For let nh meet AT m u, join nm,
and let P/ meet TA in p ; then since the tangent £n is perpen--
dicular to TR, PM is to Rr (or wN) as nh (or Ml) is toEA,
or uN ; consequently mS is to wN asPr toTr, and vm is pa-
rallel to TP. Therefore In, or T«^ is to Po as T/> is to ^, or
as T/ to TP, and the angle PTo b equal to ttn. Hence if the

VOL.!. Cc 9tQh

402 Of the Motitm rf ilm Plmnc Bookl.

arch ^ betomte micirdeas theexeessof V^G4- V above ^G — «V
» td VG-^ V, Tg be ta fcen upon TJfc in flie same ratio to TA
as •G^—2Vto^G+V,and •/parallel toT6 meet/^/perpendi-
Ottlar to Tb mf, tbe aBgle VTf shall be equal to the motion of
the nodes in this trajectory in the time the body revolves froat
Ihe quadrature B to the subsequent quadrature. When TA is
equal to TB, and the planes are sJmost coincident, ibis angle
j^'f/iB to two right ones in the rat io comp ound ed of A e ratio
of the excess of i^G+7 above v^G—^Vto ^G+V, and of
that of tfcic C98ine ofUfk to the radius nearly.
• 464. Suj^osenowthatAtelineofthenodesisinaiiysttQation
TZ' (fig^ 21 Oat the begintxingof a revolution as in art. 470.and
ih order to determme'the motion of the node during the revo-
hilion, take the arch Iq backwar ds from I in the same ra tio to
a whole circte as Ae excess of V'G+V above ^G — sV, is to
Vg+V, and fet ql parallel to OTmeet LF m /, join Tl, and
Ae trajectory sfaafimienect the plane LTS in I after a complete
revolution of the point P. For when P returns to H, XV
va nisbes, and t he arch described by Q is to a whole circle as
Vg — aV to ^G+V, by the construction. Therefore whea
F returns to H^ Q comes to q, and E to the point /in the plane
ZiTS. The angle LT^sho ws how far the nodes have moved back-*
Wardsin this time^ and is to the angle YTq as the tectangfte LGI
to the square of Tl nearly when V is much less than G. For lot
Ix be perpendicular to T/ in x, and lo parallel to AT meet ql m
Oj then Lr shall be to L/(orIo)asLG toTL^andlo toI^asIG
to TI nearly, consequently Lr is to Ij as the rectangle LGI to
the rectangle contained by TL and TI, and the an^ VTl to
Xtq as the rectangle LGI to the square of TL, ncariy. Hence
when TA is equal to.TB, V is much leas than G, and L is taken
upon the circle B4; if the ratio of V to G be given, the angle
IT; is given, and the motion of the nodes (or the angle LT/) is
as the square of LG the siile of the angle LTA the distance of
tiie node ftom the ran ; which is agreeable to what is shown
<iif the nodes of the moon by Sir liaac Newton, cor. %,prop. 30,
Bb. 3, Prindp. Supposing thereforethe motion of the nodes of
the mdon to vary in the duplicate ratio of the sine of their
^nance froift the sun, die^ annual motion of the nodes may be

- • - • briefly

nate SSSf. A .40J Wil .

t

i
f

J
i

i

i

Chap. XL * if ike Lunar Orbit. itKA

briefly oompnted fiom what was flhown ui art. 481, hjAe
hi theorem in the scholiamofprcp. SSfiib. ^^Pfindp.y
Aicated by^Mr. Machin, and ance explaioed in a mere general
manner in the laws of the nwm^$ motion, Sec. j>. 14. Bj that
tfaeoiiem, the mean motion of the «iin from the node is a geo«
metrical mean proportional between the mdion of the man an4
the mean motion of the ran from the node in the month when
the line of the nodes is in quadralwre to the sun. The mean
motion q£ the node in this month is to the motion of the sm^
according to the first proportion deduced in art. 481^ in the
i»>mponnd proportion of 1 to lUlflQiS, and of the mean mo-
tion of the moon to the mean motion of the sun, or of SlSQt^
160, that is in the prc^Kxrtioa of 1 to 9fi5S6l : oonaeqiient)^
the mean aannalmotion ofthenodeis to the motion of the son
(by the theorem) as 1 to 16,60413 ; whidi glares 19'^ fil' 9/' fer
the motion of the node in a aidereal year. Ao^^Mrdisig to the
aecond proportion in art. 46 U that was deduced ftom our con*
•truction, this motion-b about ig^ %l' ; by the principles thait
are laid down in the above mentioned treatise it is .19P 21' J" {|
and by astronomical eh a e r vation it is 1^ £l' £l" |>. In th^t
compntatiotts we have Abstracted from the acceleration of thia
motion that arises/rom the excess of the solar foroe on that half
of the moon's orbit which is towards the sun above what it ia
on the other half, and fiom any effect the exoentrieity of tho
lonar orbit, and the incveafle of the inchnationr of the plaaf
while the nodes move ftom the quadrotares to the syzjffe^
may be supposed to have upon it. But diis is soficientfor our
purpose, which is only to ilkstrate from the eodstraction <^tho
trajectory (in art. 476) the compulation of this motion from
its cause, which afibrds so remadkaUe a confirmation of the
principle of gravity, and may on oth^ accounts be of greaL
use. See the iafm ^tht wioon's motion, p. £4.

485. Ingeneralietthebody6etontfr«mtheii0doLOfg.£l£)a»
in art.4769 and let it berequired to determine die actuation o£
the line of the nodes whai it comes to any point e in the tri^:
jectory. The tangent & of the curve L£ is perpendicular
to TR,by art. 475. Let die angle TPjr be made equal to'
theangjbATH.aidPpxaeetTBiiiji; tet^ pardlel to TA

Cc8 meet

404 Of tie Motion of the Plant / "Baokl.

sneet Er in n^ joia Tn, and the nodes shall be in Ae right Use
Tn when the body comes to e after describing the arch Lc in
the tngectoiy. The dembnstralion is similar to that in art. 482,
If«N be taken npon NQin the same ratio to NQ as VG— «V
is to ^6+ V^ and mo be perpendicular to Pp in o, the angle
HTn will be equal to oTP, and LTn the motion of the node
will be either equal to the sum or difierenoe of the angles
oTP and LTH- When TP becomes perpendiculaF to TH,
Tp becomes parallel to BT> and the line of the nodes perpeft*

dicular to TR«

486. In art.479^we supposed thebody to setoutjfroiiiBinaright
line perpen<ycular to TB. But if it set out from any point L
in TB (/ig. dl3) and the right lines TB> TL with the Ydacikj
atLbe gi^n^ the motion of the node will be sUll as thestne-
oomplement of the inclination of the {dane. If TA be equal to
TB, and the arch BH.be taken from B towards th e oppo sition
a in the sameratio toasem icircle as the excess of i/G+Vabove
4/G-HiV » to v^G+V, the line of the nodes will describe an
angle equal to BTH in the time P describes the semicircle Hk;
because P and Q will come at the same time to the point Jk
This is the motion of the node that was determined in the first
part of art. 160. Supposing still the node L to be in the line
of Ihe quadratures T B, let th e ang le BTH be to a right an gle
as the diflference of ^GT7 and v'G— aV is to VqIl^Y (op
inthe caseof the moonaccording to art. 480,as 1 to 1 19,64^),
and the arch BH be taken from B towards the opposition ; let
die point P describe the circle Bb, and Q the circle LA, TA
being supposed equal to TL ; then (by art. 476) the right
fines TP and TQ will coincide with each other at the conjunct
lion, wheie the motion in the trajectory wiQ becoine peipen*
dicular to TA ; and (by art. 485), while the body moves fioni
the quadrature L to the conjunction A, the line of the nodea
will describe an angle equal to BTH. But if we suppose a bo*
ij to set out from the same point Lin a right line perpendicu*
Iqr to TL (as in art. 479), the point P to describe the drcle
IiA,and Q the circle Bb, so that ihe distance of the body from
T at the conjunction may he equal to TL its distance at the
l|nadratuve^ th§ mption of the node whilf the body movea from

thf

Chap. XL of the Lunar Orhit. 405

the quadrature to the conjunction wiUbeequal to the a ngle
ATRif the angle BTA be to VTq as v'GlTto ^G— 2V, and
yn be to Rit in the same ratio^ by art 482,, and this angle ATR
is to a right one as 1 to 121^655^6. If we suppose the motion
of the node in a circular orbit^ when the planes are almost co-
incident^ to be a mean betwixt the motion of the node in those
two cases^ it will be to the mean motion in the orbit as 1 to
1120^64273. This almost coincides with the pn^rtion that
was deduced for this case at the latter end of art. ISO, in imi«
tation of Sir Isaac Newton's method ; which in larger num«
^ben is that of 1 to 120,64275.

487* Theangle in which the planeofthelunar orbit intersects
the ecliptic is perpetually varying; and this motion may be also
illustrated from art.476. In general if EK 0(^.212)beperpen»
dicular to the line of the nodes Tn in K^ the indipation of the
plane of the trajectory to the plane BTA is an angle the taii«»
gent of which is to the radius as XV to EK, First let the line
of the nodes be in quadrature to the sun^ and suppose the body
to set out from the quadrature in a right line perpendicular lo
TB^asin art.479^ make theangle ST6equal to the inclination at
the quitdrature; takeTZ upon Tb (fig. 214 and 209) in the same
ratiotoTBasthe velocity atBin the triyectoryis to the vdod^of
P in the circle Bb; let Zd parallel to ST^ and Td perpen dicn*
k rtoSTm eetinrf; upon <fZ take (2a to ^ as v'G+V is to
VG — 2V, join Ta, and the angle STa will be nearly equal to
the inclinatioki at the syzygies* For TZ is equal to the semi*
axis that is conjugate to TB in the ellipse that would have been
described about T if the third force Tg (des cribe d in art . 472)
had not acted ; and TA is to rfZ as ^G+Vto VG— 2V: con^
sequently TA is equal to Ja. But at the syzygies the distance
of the point E from T is nearly equal to TA^ and Ee to Td.
Therefore the angle STa is the inclination at the syzygies near-
ly, and the angle aTZ is the decrement of the inclination while
the body moves from the quadratures to the syzygies. Let Zgf
be perpendicular to Ta iny, and because Zy is to Za as Ti2 is to
Ta, the angle aTZ is nearly to Tad (or STa) as Za is to TA,
that is in t he rati o compounded of that of 2a to tfa (or of the
excess of ^G+V above VG^-sV to V^§+T)andthatofrfato

T»

406 OfihMMmiofthPlane BmIeL

7* ((»r «/ tbe epwe^tke iadiiiAtioQ to Ike radm). 3«t the
■j^aao motni4Dfthf node io this revolatkm isto tbeaeannio-
tion of£wiJheittmeratio>by«rt.481« Tbecefoce (by apply-
iii|; tbJs to tbe Bioon) tht exotts of the inrltaaUoa at the q«^
dretww «ibo¥e wbait it ib at the sygygiesin thismoath is to the
letter ittoUaatieii as the Mean motien of the node in tbu aaoodl
i» te the neiean motion of the moon, or as 1 to isi^l 4 whiok
ii the retio afsigtied in the tieatise above nijentioned^ p. 4^
arh^ie the ipitrr aoedtale iodinationa are abo determined in an
ej^nnt manner. Aeooiding to this proportion the difftrenceef
the inclinations at the quadratures and sysfgiesiB about d*^"!^
•mi if Ihia difiamnoe heinci^aaed intbe ratioof the^nodical
te the ^eiiqdioel mootf^ it will then agroe nearly with thai
wUeh emelts from Sir i»^c Nemtw^a method, prop. 33> Hi. 3,
Pfincip. ifthe body be supposed tooontknieita motion in the
tn^etory, the mcti^tiQa.al the node e will be ginater than
thftimdmetiMi et the node B at the beginning of the tevohition
in the Mam miso aB the radi^
that » t» a dfde m V^G'-SV to v^G^ V.

4M. SnpposeOOif .«15)tobetheprif ectionmitheplaoeBr4
efathefdaoewhemtfaediBtanceofthebodythatdesciibmtbetnii^
jnctmy frem the piane BTA is greatest 14; the bqjinnkig of a^
iavahtfirm; aadlettbe aich OQ be taken backwards fiom Oon
the eirda OK deseiibod from the oentie T in t he sam e ratio
to ths wh de cju camfer ence as the excess ofVG+V abo^
^a*-^V is to v^GT^; ktOEparaUeltoBTmeetOfpaml.
MtDTAinfi^andTEneetthearchOQ inV; thenifTD
perpendicular to TV be equal to Qo, the angle EOV villahciir
nesdy how much the incKnelion of the pleae varies in this m*
vobtton ; because the engle TVD is equal to OTo^ and TED
is nearly equal to 4he indinaiion of the plane when the body aA
ter ahnesta eoaBplete eetolution is again attbe geeatest distance
fromBTA. When TO is wHhin the angle BTA, or 6Ta, TE
mlevthw TO, the angle TVD is less than T£D^ and cense-
flwtly the indinatioBcf thephmeinoreases: but when TO is
within the an^e KVb or aTB^TEis gieater Omn TO^ snd the-
isiciination of the plane decreases ; th^t is while the line ^i the
nodes (which is pofpeDdicular to TO) movea £rom the ^aan

dratures

J

CIiAp«lCL iff tht IJmir 0irbUi 4B9

^drafpn to tfae«j9^et^ the indinstioii of the fdawi inesoMcii;
hoi whik the line of the nodes moves fiooi the 9jtfg^
to dieqsadmtaies, the inclination decreases.

409. Let the arch Ex described from the centre D DieetDlF
in », and YG he perpendicular to TA in 6 ; then because £d^ is
to EY asTD to DY, £Y to YQ as Y6 to TO, and YQ to
OQ as- the square of TG to the square of TV nearfy, it foltMni
that ^e angle EDY is to CfTQ, in the ratio coinpoimded of
that of TD to DV, and that of the rectangle TOY to the reet-
angle contained by TY and DY. But the angle OTQ it to
the motion of the node in that month when the line of the
nodes is in quodrature to the sun (which we shaft call N) ae
DY to TV, by art. 479 ; consequently the angle EDY ia to
li in the ratio compounded of the ratio of TD to DY^ and that
of the rectongle T6Y to the square of TY^ that is in the mtio
<^ die rectangle contained by th« «im of the iaelinatioa an4
the sme of 2BTY (the double distance of the nodea from the
Mn) to twice the square of the radius; which agrees with eor.
S, pfnp. $S, la. 3, Prmeip. The decrement of the radinatioft
in a revolution wUle the nodes move from the sysffgies to diw
qoadratarss is nearly as EY; it is greatest wbm Y ib in the
oetants^ whcse EY becomes neariy equal to |0Q; and EDYy
the decrement in a revolution^ is to the moan motion of the
nodes neuiy aa TD to DY^ or as the sine of incli ne i ien to the
radma. Because the radius TO and the arch OQavesapposed to
be given^ the point E is always in an ellipde HEA> whose trana^
Terse axis is in the octants after the ^zygies; and the whole
decrement of the inclination while the nodes move from the
eyiygie^ to the quadratures is nearly to the decrement that
wonld have been generated in the same time by the variation
at the octants continued uniformly, as the area deseribed by
BY while O moves along the quadrant KA is to the rectangle
contained by ^OQ and JUc, or as the area included betwixt
the quadrants of the ellipse and circle to that rectangle, that i»
as the radius TK to the quadrant Kk. Hence the mean hour*
ly decrement of the inclination is to the mean hoarly motion of
the node as the sin^ of the inclination (the radius being TK) is
to the quadrant Kk; so that when the inclination is small, the

whole

40S Cflhtuh thai ffrwckate Book L

nrhole inelination would be generated by its mean hourly varia-*
tion in the same time that a quadrant would be described by the
nodes with theirmean motion. Therefore the wholedecrement
of the inclination while the nodes move from the syzygies to
the quadratures (or while a quadrant is generated by the nu>«
tion of the son from the node) is to thewhokinclinaUMi as the
mean motion of the node b to the mean motion of the sun from
the node^ or as 1 to 19^6 (by art. 484) ; according to which
proportion that decrement is about l6' 10". See the law$ oftinL
mooris motion^ &c. p. €6.

490. If we suppose a body to descend along the quadrant
BA 0^.2l6)> andthevelocityatBtobeequaltothevelocityofQ
as in art. 480^ the velocity at A wiUbeequal to that which would
be acquired by falling in the chord from B to A^ or by ttMng
from B to T, and then from T to A, by art. s&l. The vdodty
that would be thus acquired ^at A is the same that would be ac-
quired if the motion was not accelerated from B to T« and was
accelerated from T to A by the force Tg (fig. £05) only. Let
B6 perpendicular to TB be to Aa perpendicular to TA as
^G— 2V is to dV> and the Velocity acquired at A will be to the
velocity at B in the subduplicate ratio of the sum of the tri-
angles TBi> TAa to the triangle TB6> or of the sum of B6 and
An to 'Bb, that is as ^g+v to •g— 2V, or as v^ss^-sll to S«
In like manner the velocity at any point E is determined ; and
this will be found to agree nearly with what is shown conoerh*
ing the acceleration of the area described about the earth in a
circular orbit^ prop. £6j Uh. S, Ptincip,, where this accekralion
is computed by the increment of the velocity in such an orbit ;
and it coincides with'what is shown^ p. 29, of the la»$ of tic
moon'i motion. The same increase of the velocity may be de-
duced from what has been shown of the trajectory Be when its
planecoincides with the plane BTS (fig. 209), and itis supposed
nearly circular; for the velocity atBis equal to the velocity of Q;
when TA is equal to TB^ the velocity at the syzygies b nearly
equal to the velocity of P^ by art. 475; and these velocities are
to each other as %/g— sv to ^g+T. Some other corollaries
relating to this theoiy might be deduced from the preceding
articles.

491. If

Chap. XL towards several Centres. 409

491. If a fluid be supposed to gravitate towards two pdiits*
C and S with equal forces that are the same at all distances^ the
figure of the fluid will be an oblong spheroid^ and these two
points will be ihefoci of the generating ellipse. For supposing
AEa to be any section of the fluid through C and S^let £M and
ES{fig,2l7)he taken upon EC and ES representing the equal
ibrces towards the points C and S respectively; let MPand NQ
be perpendicular to the tangent at E in P and Q^ and £P will
be equal to EQ because of the equilibration of the fluid: the*
fluxion of CE is to the fluxion of the curve AE as EP to EM,
and the fluxion of SE to the fluxion of AE as EQ to EN^ by
prop. 17. Therefore CE and SE flow with equal motions^ and
(because SE decreases while CE increases) the sum of CE and
SE is invariable; consequently AEa is an ellipse that has itsyb-
ei in C and S. The gravity at E is to the gravity at A as the
sine of the angle CE^ is to the radius. When the forces direct-
ed towards C and S are invariable, but unequal, let D be any
point of the surface AEa that terminates the figure, and it will
appear in the same manner that the difference of CE and CD
will be to the difference of SD and SE as the force towards C
is to the force towards S. The figures by which rays of light
issuing from a given point S may be refracted so ias to
have their focus afterwards in C are of this kind. The figure
AEa is m some cases a conic section, and when it passes through
the point C or S it is a portion of an epicycloid th^ is describ*
ed when a circle revolves on an equal circle, if equal and in-
variable forces are directed towards any number of given ceiH
tres, the aggregate of the right lines drawn from any point in
the surface that terminates the fluid to those centres is invari-
able. When the forces towards the centres are inversely as
the squares of the distances, the aggregate of right lines that
are inversely as the dbtances of any point of the surface from
those centres is invariable. And the figures have an analogous
property when these forces at equal distances are not equal, but
in a given ratio. But it would be of little use in philosophy
to insist on this subject.

492. Suppose that a fluid, which gravitates towards the point
C(fig.2lS), revolves about the axisAB; let ER be perpdodieu-

VOL. I. D d )ar

410 Oftht InUfUction of the Curve ^ Booi; JU

lar to AB in R from any point E in AEB the siurfaoe of theflmd,
and let CD be the ordinate at C; then it will ^pear (as ia tfa«
last article) that the fluxion of CE shall be to the fluxion of £R
as the centrifugal force is to the gravity at E> or (because the
centrifugal force at E is to the <^ntrifugal force at D as £R is
to CD) in the compound ratio of the centrifugal force atD ta
the gravity at E, and of ER to CD« From which it follows,
that if the gravity towards C be the same at all distances^And ik
right line L be to CD as the gravity is to the centrifugal fcarce
at D| the square of ER wiU be equal to the rectangle contain-
ed by CE-— CA and 2L, which agrees with what has been
shown by Mr. Huygens. If the gravity be inversely as the
square of the distance from C^ let G represent the gravity and
y the centrifugal force at D^ and let the square of the ri^t
line Lbe to the square of CD as G+i V is to | V^ tbentbe
sdiid contained by CE and the square of ER shall be equal to
the aoUd conialaed by CSr-^A and the square of I^ and CD
will be to CA as G+ i V is to G.. But the figure of the earth
91 planets is not to be discovered by suppositions of this kind;
for as the gravity of a body results from the gravity of all ita
parts^ so the gravitation towards any body results from the
gravitation towards theparticles of which it consist3^ aa is shown
by Sit Isaac Newton; and when the figure of the fluid va-
ries from a sphere^ this gravitation is not directed towards a
fixed point. Of this we shall treat in the 14th chapter^ where
we shall show that if the earth was of an uniform density^ its
figure would be an oblate spheroid accurately,

493. We have insisted at so great length on what relates to
the curvature of lines, and to the application of this theory to
philosophical problems, because in this consists one. of the
greatest advantages of the modem geometry. There are many
other problems that depend on this theory, but we shall con-
clude this subject with observing, that as when a right line in-
tersects an arch of a curve in two points, if, by varying the po-
sition of that line, the two intersections unite in one point, it
tlien becomes the tangent of the arch; so when a circle touches
a curve in one point, and intersects it in another, if, by varying
the centre, this intersection join the point of contact^ the circle

then

Chap. XL UMi Ciftk of CunahiH. 411

tlien hai the dosett contact with the arch and hecames the

circfe of cimratoie \ but it still continues to intersect the Havrt

at the same point where it touches it (that is where the same

lightUneis their common tangent, according to thedefiniticmin

art. 181), unless another intersection join that point at the same

time. Let MEm (fig* 219) be any arch that is oontinned

from E on both sides of the ray of curvature CE> ACB

a right line perpendicular to C£ ; let the right Une CM

revolve about C^ and meet the curve in M^ and PM be

perpendicular to AB in P; then if AP be supposed to

flow uniformly, the first and second fluxions of CM vanish

when M comes to E ; and, when the third fluxion of CM,

or of PM^ does not vanish at the same time^ the circle of curva-

ture intersepta the curv^ ^t £« In generalf^hen M comes to Ey

and the number o( the fluxions of CM of successive orders (in*

eluding the fii^st fluxion) which then vanish is even, the circle of

curvature atE intersects the arch Ml^mf.butlf ibU bo wiodd

number, there is no such intersection at E. For lei PN betaken

upon PM always equal to CM ; then the. curve NEn described

by N will pass through.E ; and it is manifest;, that the circle of

curvature intersects the arch MEm only when NEitintersects its

tan^nt at H, that is when, NEji has a point pf contrary flexure

at E. Therefore (by art« 266), the circle of curvature at £ in^

tersects the arch M£m only when the number of those fluxions

of CM that vanish at the term of the time when M copies to £

is an even number. This theorem may serve to illustrate

a subject that was disputed some time ago by two t;e)ebrated

authors. One of them imagined that two points of contact,

or four intersections, of the curve and circle of curvatuse must

Join each other to form an osculation. But Mr. Jaime* Bcr^^

nomlU insisted, on just grounds, that the coalition of one

point of contact and one intersection, or of three intersections^

was sufficient. In which case (and in general when an odd num*

ber of intersections only join each other) the point where they

coincide continues to be an intersection of the curve and circfe

of curvature^ as well as a point of their mutual contact and

osculation.

D d ft 404. Lt

412 Rnhark$ on the preceding Part. Book I.

494* In tbc preceding chapters we deduced ine principal
propositions.of the method of fluxions, and those upon which
its application to the resolution of problems depends, from the
plain axioms concerning ^lotion, in art. 15. Having proposed
this doctrine (art. 2), as a method of deducing the relations
of quantities, by comparing the motions which are concaved to
generate them, it seemed natural to establish this method on the
most evident principlies that relate to motion; andif it be noless
known to us than extension, it would seem,* that as this doc*
trine is far more general and comprehensive than the common
geometry, so it cannot be said to be inferior to it in accuracy
or evidence. Nor can it seem improper to dednce the propor-
ties of figures from the same principles which serve for describ-
ing their genesis. Thus, the definition of a fliixion (art. 1 1),
seemed naturally to lead us into themethodof treating^isdoc-'
trine which we have Jhllowcd' hitherto ; and its connexion
with the manner in which the genesis of figures is most com*
monly described in g^mctry, and with the most useful theories
concerning motion in philosophy (of which we have had sdmc
examples in this chapter^ induced us the rather to puisne it.
But we have insisted on it at so great length, chiefly, because
afuUaccount of the manner in which thf! principal proportions
of the method of fluxions are demonstrated by it may be of
use for removing several objections that have been lately urged
against this doctrine, whichhasbecnrepresented as depending
on nice and intricate notions; while ft has been insinuated, that
they who have treated of it have been earnest talher to go on
fast and far, than sblicitous to set out warily, and see their Way
distinctly. But we now proceed to the more concise methocfa
by 'which the fluxions of quantities are usually determined.

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